From 514f647de89e9f48272135ef234aa719b6cffbca Mon Sep 17 00:00:00 2001 From: James Mitchell Date: Fri, 30 Oct 2026 16:41:25 +0000 Subject: [PATCH 1/5] Final m11 --- docs/.nav.yml | 9 +- docs/examples/atlas/mathieu/m11.md | 112 ----------------- docs/examples/atlas/mathieu/m11/11.md | 129 ++++++++++++++++++++ docs/examples/atlas/mathieu/m11/12.md | 128 +++++++++++++++++++ docs/examples/atlas/mathieu/m11/165.md | 128 +++++++++++++++++++ docs/examples/atlas/mathieu/m11/55.md | 129 ++++++++++++++++++++ docs/examples/atlas/mathieu/m11/66.md | 128 +++++++++++++++++++ docs/examples/atlas/mathieu/m11/7920.md | 118 ++++++++++++++++++ docs/examples/atlas/mathieu/m11/index.md | 40 ++++++ docs/examples/atlas/mathieu/m11/m11-11.mp4 | Bin 0 -> 165299 bytes docs/examples/atlas/mathieu/m11/m11-12.mp4 | Bin 0 -> 169424 bytes docs/examples/atlas/mathieu/m11/m11-165.mp4 | Bin 0 -> 160120 bytes docs/examples/atlas/mathieu/m11/m11-55.mp4 | Bin 0 -> 135809 bytes docs/examples/atlas/mathieu/m11/m11-66.mp4 | Bin 0 -> 146469 bytes docs/examples/atlas/mathieu/m11/m11.mp4 | Bin 0 -> 244629 bytes 15 files changed, 808 insertions(+), 113 deletions(-) delete mode 100644 docs/examples/atlas/mathieu/m11.md create mode 100644 docs/examples/atlas/mathieu/m11/11.md create mode 100644 docs/examples/atlas/mathieu/m11/12.md create mode 100644 docs/examples/atlas/mathieu/m11/165.md create mode 100644 docs/examples/atlas/mathieu/m11/55.md create mode 100644 docs/examples/atlas/mathieu/m11/66.md create mode 100644 docs/examples/atlas/mathieu/m11/7920.md create mode 100644 docs/examples/atlas/mathieu/m11/index.md create mode 100644 docs/examples/atlas/mathieu/m11/m11-11.mp4 create mode 100644 docs/examples/atlas/mathieu/m11/m11-12.mp4 create mode 100644 docs/examples/atlas/mathieu/m11/m11-165.mp4 create mode 100644 docs/examples/atlas/mathieu/m11/m11-55.mp4 create mode 100644 docs/examples/atlas/mathieu/m11/m11-66.mp4 create mode 100644 docs/examples/atlas/mathieu/m11/m11.mp4 diff --git a/docs/.nav.yml b/docs/.nav.yml index 6be8464..0855d47 100644 --- a/docs/.nav.yml +++ b/docs/.nav.yml @@ -20,7 +20,14 @@ nav: - "Verifying presentations for sporadic simple groups": - examples/atlas/index.md - "Mathieu groups": - - "Mathieu group M11": examples/atlas/mathieu/m11.md + - "Mathieu group M11": + - examples/atlas/mathieu/m11/index.md + - "Trivial": examples/atlas/mathieu/m11/7920.md + - "M10 = A6.2_3": examples/atlas/mathieu/m11/11.md + - "L2(11)": examples/atlas/mathieu/m11/12.md + - "M9:2": examples/atlas/mathieu/m11/55.md + - "S5": examples/atlas/mathieu/m11/66.md + - "2S4": examples/atlas/mathieu/m11/165.md - "Mathieu group M12": examples/atlas/mathieu/m12.md - "Mathieu group M22": examples/atlas/mathieu/m22.md - "Mathieu group M23": diff --git a/docs/examples/atlas/mathieu/m11.md b/docs/examples/atlas/mathieu/m11.md deleted file mode 100644 index 50a9f94..0000000 --- a/docs/examples/atlas/mathieu/m11.md +++ /dev/null @@ -1,112 +0,0 @@ -# Mathieu group M~11~ - -The claims on this page come from -[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). - -## Claim - -$$ - \mathrm{M}_{11} = \langle a, b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^6 = ababab^{−1}abab^2ab^{−1}abab^{−1}ab^{−1} = 1 \rangle -$$ - -with - -$$ - |\mathrm{M}_{11}| = 7,920. -$$ - -On this page, we verify that the above claimed presentation of the Mathieu group -M~11~ defines a group of order $7,920$. - -## The code - -In [libsemigroups_pybind11][], the following script constructs the presentation -for M~11~ and runs the Todd-Coxeter algorithm. - -
-Code - -```python -from libsemigroups_pybind11 import ( - Presentation, - ToddCoxeter, - congruence_kind, - presentation, -) -from libsemigroups_pybind11.words import parse_relations - -# Setup the presentation object with the empty word and inverses, so it can represent a group -p = Presentation("abAB") -p.contains_empty_word(True) -presentation.add_inverse_rules(p, "ABab") - -# Add the defining relations -presentation.add_rule(p, parse_relations("a^2"), "") -presentation.add_rule(p, parse_relations("b^4"), "") -presentation.add_rule(p, parse_relations("(ab)^11"), "") -presentation.add_rule(p, parse_relations("(ab^2)^6"), "") -presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") - -# Run the Todd-Coxeter algorithm -tc = ToddCoxeter(congruence_kind.twosided, p) -tc.run() - -print(f"The size of the group is {tc.number_of_classes()}") -``` - -
- -## The output - -Running the above script produces the following output: - -``` -++++++++++++++++++++++++++++++++ -#0: ToddCoxeter: running for approx. 10.000s -++++++++++++++++++++++++++++++++ -#0: ToddCoxeter: RUN 0 START (strategy() = hlt) -#0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 -++++++++++++++++++++++++++++++++ -#0: ToddCoxeter: HLT 0.0 START -#0: ToddCoxeter: HLT 0.0.0 | active | killed | defined -#0: ToddCoxeter: nodes | 1 | 0 | 1 -#0: ToddCoxeter: | active | missing | % complete -#0: ToddCoxeter: edges | 0 | 4 | 0.0% -#0: ToddCoxeter: time | run 0 = 19µs | all runs = 19µs | elapsed = 641µs -++++++++++++++++++++++++++++++++ -#0: ToddCoxeter: HLT 0.0 STOP -#0: ToddCoxeter: HLT 0.0.1 | active | killed | defined -#0: ToddCoxeter: nodes | 7,920 | 1,992,427 | 2,000,347 -#0: ToddCoxeter: diff 0.0.0 | +7,919 | +1,992,427 | +2,000,346 -#0: ToddCoxeter: | active | missing | % complete -#0: ToddCoxeter: edges | 31,680 | 0 | 100.0% -#0: ToddCoxeter: diff 0.0.0 | +31,680 | -4 | +100.0% -#0: ToddCoxeter: phase 0.0 = 209ms | run 0 = 209ms | all runs = 209ms | elapsed = 210ms -++++++++++++++++++++++++++++++++ -#0: ToddCoxeter: HLT 0.1.2 | active | killed | defined -#0: ToddCoxeter: nodes | 7,920 | 1,992,427 | 2,000,347 -#0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0 -#0: ToddCoxeter: diff 0.1.0 | +7,919 | +1,992,427 | +2,000,346 -#0: ToddCoxeter: | active | missing | % complete -#0: ToddCoxeter: edges | 31,680 | 0 | 100.0% -#0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0.0% -#0: ToddCoxeter: diff 0.1.0 | +31,680 | -4 | +100.0% -#0: ToddCoxeter: phase 0.1 = 209ms | run 0 = 209ms | all runs = 209ms | elapsed = 210ms -++++++++++++++++++++++++++++++++ -#0: ToddCoxeter: RUN 0 STOP (finished) -#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch -#0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 -#0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 209ms (100%) | - (0%) -#0: ToddCoxeter: phase 0.1 = 209ms | run 0 = 209ms | all runs = 209ms | elapsed = 210ms -The size of the group is 7920 -``` - -:material-checkbox-marked-circle-outline: The computed size of the group matches -the size of the group provided on the ATLAS: $7,920$. - -!!! note - - The output you see when you run the script yourself might be different from - the above; the numbers produced and time taken will depend on the machine - you are using. However, the size of the group that is reported should be - the same as above. diff --git a/docs/examples/atlas/mathieu/m11/11.md b/docs/examples/atlas/mathieu/m11/11.md new file mode 100644 index 0000000..dc94073 --- /dev/null +++ b/docs/examples/atlas/mathieu/m11/11.md @@ -0,0 +1,129 @@ +# $\mathrm{M}_{10} = A_6.2_3$ as a subgroup of Mathieu group M~11~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max1W1). + +## The computation + +The recorded run took about 14 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +If + +$$ + H = \langle (ab)^{-4}a(ab)^4,\ (ab^2)^{-2}(abababbab)(ab^2)^2 \rangle \cong \mathrm{M}_{10} = A_6.2_3, +$$ + +then + +$$ + [\mathrm{M}_{11}:H] = 11. +$$ + +We verify the claimed index, but not the claimed isomorphism type. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") + +# M10 = A6.2_3, order 720, expected index 11. +# Generators: (ab)^-4 a (ab)^4, (ab^2)^-2 (abababbab) (ab^2)^2. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max1W1 +# Upper-case letters denote inverses. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ("(BA)^4a(ab)^4", "(BBA)^2abababbab(abb)^2"): + tc.add_generating_pair(parse_relations(word), "") + +tc.report_every(timedelta(milliseconds=0.1)) + +actual_index = tc.number_of_classes() +print(f"The index of M10 is {actual_index}") +if actual_index != 11: + raise RuntimeError(f"expected index 11, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 36 | 0 | 36 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 37 | 107 | 25.7% + #0: ToddCoxeter: time | run 0 = 29µs | all runs = 29µs | elapsed = 223µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 44 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.101 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +7 | -107 | +74.3% + #1: ToddCoxeter: phase 0.1 = 14ms | run 0 = 14ms | all runs = 14ms | elapsed = 14ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 13ms (92%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 14ms | run 0 = 14ms | all runs = 14ms | elapsed = 14ms + The index of M10 is 11 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $11$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m11/12.md b/docs/examples/atlas/mathieu/m11/12.md new file mode 100644 index 0000000..25bf769 --- /dev/null +++ b/docs/examples/atlas/mathieu/m11/12.md @@ -0,0 +1,128 @@ +# $L_2(11)$ as a subgroup of Mathieu group M~11~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max2W1). + +## The computation + +The recorded run took about 1 millisecond. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +If + +$$ + H = \langle a,\ (bab)^2 \rangle \cong L_2(11), +$$ + +then + +$$ + [\mathrm{M}_{11}:H] = 12. +$$ + +We verify the claimed index, but not the claimed isomorphism type. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") + +# L2(11), order 660, expected index 12. +# Generators: a, (bab)^2. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max2W1 +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ("a", "babbab"): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=1)) + +actual_index = tc.number_of_classes() +print(f"The index of L2(11) is {actual_index}") +if actual_index != 12: + raise RuntimeError(f"expected index 12, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 + ++++++++++++++++++++++++++++++++ + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 6 | 0 | 6 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 7 | 17 | 29.2% + #0: ToddCoxeter: time | run 0 = 31µs | all runs = 31µs | elapsed = 91µs + #1: ToddCoxeter: HLT 0.0.0 | active | killed | defined + [... lines omitted ...] + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 983µs (91%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 1ms | run 0 = 1ms | all runs = 1ms | elapsed = 1ms + #1: ToddCoxeter: HLT 0.1.57 | active | killed | defined + #1: ToddCoxeter: nodes | 12 | 27,530 | 27,542 + #1: ToddCoxeter: diff 0.1.56 | +0 | +0 | +0 + #1: ToddCoxeter: diff 0.1.0 | +6 | +27,530 | +27,536 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 48 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.56 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +41 | -17 | +70.8% + #1: ToddCoxeter: phase 0.1 = 1ms | run 0 = 1ms | all runs = 1ms | elapsed = 1ms + The index of L2(11) is 12 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $12$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m11/165.md b/docs/examples/atlas/mathieu/m11/165.md new file mode 100644 index 0000000..29d90c2 --- /dev/null +++ b/docs/examples/atlas/mathieu/m11/165.md @@ -0,0 +1,128 @@ +# $2S_4$ as a subgroup of Mathieu group M~11~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max5W1). + +## The computation + +The recorded run took about 1 millisecond. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +If + +$$ + H = \langle a,\ (babab)^2 \rangle \cong 2S_4, +$$ + +then + +$$ + [\mathrm{M}_{11}:H] = 165. +$$ + +We verify the claimed index, but not the claimed isomorphism type. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") + +# 2S4, order 48, expected index 165. +# Generators: a, (babab)^2. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max5W1 +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ("a", "babab" * 2): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=1)) + +actual_index = tc.number_of_classes() +print(f"The index of 2S4 is {actual_index}") +if actual_index != 165: + raise RuntimeError(f"expected index 165, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 + ++++++++++++++++++++++++++++++++ + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 10 | 0 | 10 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 11 | 29 | 27.5% + #0: ToddCoxeter: time | run 0 = 33µs | all runs = 33µs | elapsed = 91µs + #1: ToddCoxeter: HLT 0.0.0 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: diff 0.1.45 | +0 | +0 | +0 + #1: ToddCoxeter: diff 0.1.0 | +155 | +23,936 | +24,091 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 660 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.45 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +649 | -29 | +72.5% + #1: ToddCoxeter: phase 0.1 = 982µs | run 0 = 1ms | all runs = 1ms | elapsed = 1ms + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 885µs (88%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 990µs | run 0 = 1ms | all runs = 1ms | elapsed = 1ms + The index of 2S4 is 165 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $165$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m11/55.md b/docs/examples/atlas/mathieu/m11/55.md new file mode 100644 index 0000000..4f4a4ca --- /dev/null +++ b/docs/examples/atlas/mathieu/m11/55.md @@ -0,0 +1,129 @@ +# $\mathrm{M}_9:2$ as a subgroup of Mathieu group M~11~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max3W1). + +## The computation + +The recorded run took about 1 millisecond. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +If + +$$ + H = \langle (ab)^{-2}a(ab)^2,\ (ab^2)^{-1}b(ab^2) \rangle \cong \mathrm{M}_9:2, +$$ + +then + +$$ + [\mathrm{M}_{11}:H] = 55. +$$ + +We verify the claimed index, but not the claimed isomorphism type. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") + +# M9:2, order 144, expected index 55. +# Generators: (ab)^-2 a (ab)^2, (ab^2)^-1 b (ab^2). +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max3W1 +# Upper-case letters denote inverses. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ("(BA)^2a(ab)^2", "BBAbabb"): + tc.add_generating_pair(parse_relations(word), "") + +tc.report_every(timedelta(microseconds=1)) + +actual_index = tc.number_of_classes() +print(f"The index of M9:2 is {actual_index}") +if actual_index != 55: + raise RuntimeError(f"expected index 55, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 + ++++++++++++++++++++++++++++++++ + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #1: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #1: ToddCoxeter: nodes | 14 | 0 | 14 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 15 | 41 | 26.8% + #1: ToddCoxeter: time | run 0 = 42µs | all runs = 42µs | elapsed = 298µs + ++++++++++++++++++++++++++++++++ + [... lines omitted ...] + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 1ms (88%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 1ms | run 0 = 1ms | all runs = 1ms | elapsed = 1ms + #1: ToddCoxeter: HLT 0.1.48 | active | killed | defined + #1: ToddCoxeter: nodes | 55 | 28,069 | 28,124 + #1: ToddCoxeter: diff 0.1.47 | +0 | +0 | +0 + #1: ToddCoxeter: diff 0.1.0 | +41 | +28,069 | +28,110 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 220 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.47 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +205 | -41 | +73.2% + #1: ToddCoxeter: phase 0.1 = 1ms | run 0 = 1ms | all runs = 1ms | elapsed = 1ms + The index of M9:2 is 55 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $55$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m11/66.md b/docs/examples/atlas/mathieu/m11/66.md new file mode 100644 index 0000000..ad9512c --- /dev/null +++ b/docs/examples/atlas/mathieu/m11/66.md @@ -0,0 +1,128 @@ +# $S_5$ as a subgroup of Mathieu group M~11~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max4W1). + +## The computation + +The recorded run took about 6 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +If + +$$ + H = \langle a,\ (ab^2ab^2ab)^2 \rangle \cong S_5, +$$ + +then + +$$ + [\mathrm{M}_{11}:H] = 66. +$$ + +We verify the claimed index, but not the claimed isomorphism type. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") + +# S5, order 120, expected index 66. +# Generators: a, (abbabbab)^2. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M11/words/M11G1-max4W1 +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ("a", "abbabbab" * 2): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=35)) + +actual_index = tc.number_of_classes() +print(f"The index of S5 is {actual_index}") +if actual_index != 66: + raise RuntimeError(f"expected index 66, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 15 | 0 | 15 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 16 | 44 | 26.7% + #0: ToddCoxeter: time | run 0 = 26µs | all runs = 26µs | elapsed = 85µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 264 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.99 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +248 | -44 | +73.3% + #1: ToddCoxeter: phase 0.1 = 6ms | run 0 = 6ms | all runs = 6ms | elapsed = 6ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 5ms (90%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 6ms | run 0 = 6ms | all runs = 6ms | elapsed = 6ms + The index of S5 is 66 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $66$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m11/7920.md b/docs/examples/atlas/mathieu/m11/7920.md new file mode 100644 index 0000000..258ec8c --- /dev/null +++ b/docs/examples/atlas/mathieu/m11/7920.md @@ -0,0 +1,118 @@ +# The trivial subgroup of the Mathieu group M~11~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). +Enumerating over the trivial subgroup computes the order of the group. + +## The computation + +The recorded run took about 108 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $H$ be the trivial subgroup. We verify that the +[presentation of M~11~](index.md#presentation) defines a group of order + +$$ + [\mathrm{M}_{11}:H] = |\mathrm{M}_{11}| = 7,920. +$$ + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^6"), "") +presentation.add_rule(p, parse_relations("ababaBabab^2aBabaBaB"), "") + +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.report_every(timedelta(milliseconds=1)) +tc.run_for(timedelta(seconds=10)) + +if not tc.finished(): + raise RuntimeError("The enumeration did not finish within 10 seconds") + +actual_order = tc.number_of_classes() +print(f"The size of the group defined by the presentation is {actual_order}") +if actual_order != 7920: + raise RuntimeError(f"expected order 7920, got {actual_order}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: running for approx. 10.000s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 22], ∑(|u| + |v|) = 73 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 4 | 0.0% + #0: ToddCoxeter: time | run 0 = 13µs | all runs = 13µs | elapsed = 66µs + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 31,680 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.85 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +31,680 | -4 | +100.0% + #1: ToddCoxeter: phase 0.1 = 106ms | run 0 = 106ms | all runs = 106ms | elapsed = 106ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 100ms (93%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 108ms | run 0 = 108ms | all runs = 108ms | elapsed = 108ms + The size of the group defined by the presentation is 7920 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed order is the claimed order: $7,920$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m11/index.md b/docs/examples/atlas/mathieu/m11/index.md new file mode 100644 index 0000000..8edecf1 --- /dev/null +++ b/docs/examples/atlas/mathieu/m11/index.md @@ -0,0 +1,40 @@ +# Mathieu group M~11~ + + + +The presentation and maximal subgroup data come from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M11/). + +## Presentation + +$$ + \mathrm{M}_{11} = \langle a,b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^6 = ababab^{-1}abab^2ab^{-1}abab^{-1}ab^{-1} = 1 \rangle +$$ + +with $|\mathrm{M}_{11}| = 7,920$. + +## Subgroup enumerations + +The following table summarises the enumeration over the trivial subgroup and +the enumerations over all five maximal subgroups. Each linked page includes +self-contained code, recorded output, and an animation of the corresponding +computation. + +
+ +| Subgroup | Subgroup order | Index in $\mathrm{M}_{11}$ | Verification | +| :------- | -------------: | ------------------------: | :----------: | +| [Trivial](7920.md) | $1$ | $7,920$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$\mathrm{M}_{10} = A_6.2_3$](11.md) | $720$ | $11$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$L_2(11)$](12.md) | $660$ | $12$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$\mathrm{M}_9:2$](55.md) | $144$ | $55$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$S_5$](66.md) | $120$ | $66$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$2S_4$](165.md) | $48$ | $165$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | + +**Key** + +| Symbol | Meaning | +| :----: | :------ | +| :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | We could verify the index. | + +
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-Code - -```python -from libsemigroups_pybind11 import ( - Presentation, - ToddCoxeter, - congruence_kind, - presentation, -) -from libsemigroups_pybind11.words import parse_relations - -# Setup the presentation object with the empty word and inverses, so it can represent a group -p = Presentation("abAB") -p.contains_empty_word(True) -presentation.add_inverse_rules(p, "ABab") - -# Add the defining relations -presentation.add_rule(p, parse_relations("a^2"), "") -presentation.add_rule(p, parse_relations("b^3"), "") -presentation.add_rule(p, parse_relations("(ab)^11"), "") -presentation.add_rule(p, parse_relations("(a,b)^6"), "") -presentation.add_rule(p, parse_relations("(ababaB)^6"), "") - - -# Run the Todd-Coxeter algorithm -tc = ToddCoxeter(congruence_kind.twosided, p) -tc.run() - -print(f"The size of the group is {tc.number_of_classes()}") -``` - -
- -## The output - -Running the above script produces the following output: - -??? info "Output from the Python script" - ``` - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: running for approx. 10.000s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 START (strategy() = hlt) - #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 START - #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined - #0: ToddCoxeter: nodes | 1 | 0 | 1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 0 | 4 | 0.0% - #0: ToddCoxeter: time | run 0 = 21µs | all runs = 21µs | elapsed = 103µs - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 STOP - #0: ToddCoxeter: HLT 0.0.1 | active | killed | defined - #0: ToddCoxeter: nodes | 5,000,004 | 2,964,327 | 7,964,331 - #0: ToddCoxeter: diff 0.0.0 | +5,000,003 | +2,964,327 | +7,964,330 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 5,782,533 | 14,217,483 | 28.9% - #0: ToddCoxeter: diff 0.0.0 | +5,782,533 | +14,217,479 | +28.9% - #0: ToddCoxeter: phase 0.0 = 624ms | run 0 = 624ms | all runs = 624ms | elapsed = 624ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined - #0: ToddCoxeter: nodes | 5,000,004 | 2,964,327 | 7,964,331 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 5,782,533 | 14,217,483 | 28.9% - #0: ToddCoxeter: time | run 0 = 624ms | all runs = 624ms | elapsed = 624ms - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 5,000,004 - #0: ToddCoxeter: n = lookahead_next() = 5,000,000 - #0: ToddCoxeter: large collapse, number of coincidences 100,002 >= 100,000 = large_collapse()! - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined - #1: ToddCoxeter: nodes | 1,909,907 | 6,057,700 | 7,964,331 - #1: ToddCoxeter: diff 0.1.0 | -3,090,097 | +3,093,373 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 0 | 7,639,628 | 0.0% - #1: ToddCoxeter: diff 0.1.0 | -5,782,533 | -6,577,855 | -28.9% - #1: ToddCoxeter: phase 0.1 = 375ms | run 0 = 1.001s | all runs = 1.001s | elapsed = 1.001s - #1: ToddCoxeter: lookahead progress | ~32.6% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.1 STOP - #0: ToddCoxeter: LOOKAHEAD 0.1.2 | active | killed | defined - #0: ToddCoxeter: nodes | 95,046 | 7,869,285 | 7,964,331 - #0: ToddCoxeter: diff 0.1.1 | -1,814,861 | +1,811,585 | +0 - #0: ToddCoxeter: diff 0.1.0 | -4,904,958 | +4,904,958 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 380,177 | 7 | 100.0% - #0: ToddCoxeter: diff 0.1.1 | +380,177 | -7,639,621 | +100.0% - #0: ToddCoxeter: diff 0.1.0 | -5,402,356 | -14,217,476 | +71.1% - #0: ToddCoxeter: phase 0.1 = 491ms | run 0 = 1.116s | all runs = 1.116s | elapsed = 1.116s - #0: ToddCoxeter: lookahead_next() is now max(f x a = 190,092, m = 10,000) (-4,809,908) - #0: ToddCoxeter: because f x a < n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 95,046 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: m = lookahead_min() = 10,000 - #0: ToddCoxeter: n = lookahead_next() = 5,000,000 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.2 START - #0: ToddCoxeter: HLT 0.2.0 | active | killed | defined - #0: ToddCoxeter: nodes | 95,046 | 7,869,285 | 7,964,331 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 380,177 | 7 | 100.0% - #0: ToddCoxeter: time | run 0 = 1.116s | all runs = 1.116s | elapsed = 1.116s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.2 STOP - #0: ToddCoxeter: HLT 0.2.1 | active | killed | defined - #0: ToddCoxeter: nodes | 95,040 | 7,869,292 | 7,964,332 - #0: ToddCoxeter: diff 0.2.0 | -6 | +7 | +1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 380,160 | 0 | 100.0% - #0: ToddCoxeter: diff 0.2.0 | -17 | -7 | +0.0% - #0: ToddCoxeter: phase 0.2 = 19ms | run 0 = 1.136s | all runs = 1.136s | elapsed = 1.136s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.3.2 | active | killed | defined - #0: ToddCoxeter: nodes | 95,040 | 7,869,292 | 7,964,332 - #0: ToddCoxeter: diff 0.3.1 | +0 | +0 | +0 - #0: ToddCoxeter: diff 0.3.0 | -6 | +7 | +1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 380,160 | 0 | 100.0% - #0: ToddCoxeter: diff 0.3.1 | +0 | +0 | +0.0% - #0: ToddCoxeter: diff 0.3.0 | -17 | -7 | +0.0% - #0: ToddCoxeter: phase 0.3 = 19ms | run 0 = 1.136s | all runs = 1.136s | elapsed = 1.136s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 STOP (finished) - #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch - #0: ToddCoxeter: num. phases | 1 | 0 | 2 | 0 - #0: ToddCoxeter: time spent in phases | 491ms (43%) | - (0%) | 644ms (57%) | - (0%) - #0: ToddCoxeter: phase 0.3 = 20ms | run 0 = 1.137s | all runs = 1.137s | elapsed = 1.137s - The size of the group is 95040 - ``` - -:material-checkbox-marked-circle-outline: The computed size of the group matches -the size of the group provided on the ATLAS: $95,040$. diff --git a/docs/examples/atlas/mathieu/m12/12-i.md b/docs/examples/atlas/mathieu/m12/12-i.md new file mode 100644 index 0000000..bd24516 --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/12-i.md @@ -0,0 +1,136 @@ +# $\mathrm{M}_{11}$ (class I) as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max1W1). + +## The computation + +The recorded run took about 136 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below, +for class I. If + +$$ + H = \langle u,v \rangle \cong \mathrm{M}_{11}, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 12. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# M11 (class i), order 7920, expected index 12. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max1W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "BaBabab", + "baBababaBaBabaBabababaBaBabaBaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=1000)) + +actual_index = tc.number_of_classes() +print(f"The index of M11 (class i) is {actual_index}") +if actual_index != 12: + raise RuntimeError(f"expected index 12, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 39 | 0 | 39 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 40 | 116 | 25.6% + #0: ToddCoxeter: time | run 0 = 29µs | all runs = 29µs | elapsed = 84µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 48 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.109 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +8 | -116 | +74.4% + #1: ToddCoxeter: phase 0.1 = 135ms | run 0 = 136ms | all runs = 136ms | elapsed = 136ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 133ms (97%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 136ms | run 0 = 136ms | all runs = 136ms | elapsed = 136ms + The index of M11 (class i) is 12 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $12$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/12-ii.md b/docs/examples/atlas/mathieu/m12/12-ii.md new file mode 100644 index 0000000..164e434 --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/12-ii.md @@ -0,0 +1,136 @@ +# $\mathrm{M}_{11}$ (class II) as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max2W1). + +## The computation + +The recorded run took about 157 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below, +for class II. If + +$$ + H = \langle u,v \rangle \cong \mathrm{M}_{11}, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 12. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# M11 (class ii), order 7920, expected index 12. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max2W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "babaBababaBabaBaBabababaBabaBaBaBaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=1200)) + +actual_index = tc.number_of_classes() +print(f"The index of M11 (class ii) is {actual_index}") +if actual_index != 12: + raise RuntimeError(f"expected index 12, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 37 | 0 | 37 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 38 | 110 | 25.7% + #0: ToddCoxeter: time | run 0 = 26µs | all runs = 26µs | elapsed = 76µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 150ms (95%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 157ms | run 0 = 157ms | all runs = 157ms | elapsed = 157ms + #1: ToddCoxeter: HLT 0.1.106 | active | killed | defined + #1: ToddCoxeter: nodes | 12 | 1,746,744 | 1,746,756 + #1: ToddCoxeter: diff 0.1.105 | +0 | +0 | +0 + #1: ToddCoxeter: diff 0.1.0 | -25 | +1,746,744 | +1,746,719 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 48 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.105 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +10 | -110 | +74.3% + #1: ToddCoxeter: phase 0.1 = 157ms | run 0 = 157ms | all runs = 157ms | elapsed = 157ms + The index of M11 (class ii) is 12 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $12$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/1320.md b/docs/examples/atlas/mathieu/m12/1320.md new file mode 100644 index 0000000..6fd24dd --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/1320.md @@ -0,0 +1,135 @@ +# $A_4 \times S_3$ as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max11W1). + +## The computation + +The recorded run took about 59 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong A_4 \times S_3, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 1,320. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# A4 x S3, order 72, expected index 1320. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max11W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "abaBababaBababaBab", + "babaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=500)) + +actual_index = tc.number_of_classes() +print(f"The index of A4 x S3 is {actual_index}") +if actual_index != 1320: + raise RuntimeError(f"expected index 1320, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 22 | 0 | 22 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 23 | 65 | 26.1% + #0: ToddCoxeter: time | run 0 = 27µs | all runs = 27µs | elapsed = 82µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 5,280 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.93 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +5,257 | -65 | +73.9% + #1: ToddCoxeter: phase 0.1 = 59ms | run 0 = 59ms | all runs = 59ms | elapsed = 59ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 56ms (94%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 59ms | run 0 = 59ms | all runs = 59ms | elapsed = 59ms + The index of A4 x S3 is 1320 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $1,320$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/144.md b/docs/examples/atlas/mathieu/m12/144.md new file mode 100644 index 0000000..55720d1 --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/144.md @@ -0,0 +1,135 @@ +# $L_2(11)$ as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max5W1). + +## The computation + +The recorded run took about 99 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong L_2(11), +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 144. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# L2(11), order 660, expected index 144. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max5W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "aBababaBababaBabab", + "bababaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=800)) + +actual_index = tc.number_of_classes() +print(f"The index of L2(11) is {actual_index}") +if actual_index != 144: + raise RuntimeError(f"expected index 144, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 26 | 0 | 26 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 27 | 77 | 26.0% + #0: ToddCoxeter: time | run 0 = 26µs | all runs = 26µs | elapsed = 77µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 576 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.98 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +549 | -77 | +74.0% + #1: ToddCoxeter: phase 0.1 = 99ms | run 0 = 99ms | all runs = 99ms | elapsed = 99ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 92ms (93%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 99ms | run 0 = 99ms | all runs = 99ms | elapsed = 99ms + The index of L2(11) is 144 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $144$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/220-i.md b/docs/examples/atlas/mathieu/m12/220-i.md new file mode 100644 index 0000000..baa18bb --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/220-i.md @@ -0,0 +1,136 @@ +# $3^2:2S_4$ (class I) as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max6W1). + +## The computation + +The recorded run took about 17 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below, +for class I. If + +$$ + H = \langle u,v \rangle \cong 3^2:2S_4, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 220. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# 3^2:2S4 (class i), order 432, expected index 220. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max6W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "Bab", + "babababaBaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=150)) + +actual_index = tc.number_of_classes() +print(f"The index of 3^2:2S4 (class i) is {actual_index}") +if actual_index != 220: + raise RuntimeError(f"expected index 220, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 15 | 0 | 15 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 16 | 44 | 26.7% + #0: ToddCoxeter: time | run 0 = 28µs | all runs = 28µs | elapsed = 80µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 880 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.88 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +864 | -44 | +73.3% + #1: ToddCoxeter: phase 0.1 = 17ms | run 0 = 17ms | all runs = 17ms | elapsed = 17ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 15ms (90%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 17ms | run 0 = 17ms | all runs = 17ms | elapsed = 17ms + The index of 3^2:2S4 (class i) is 220 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $220$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/220-ii.md b/docs/examples/atlas/mathieu/m12/220-ii.md new file mode 100644 index 0000000..d86d585 --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/220-ii.md @@ -0,0 +1,136 @@ +# $3^2:2S_4$ (class II) as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max7W1). + +## The computation + +The recorded run took about 45 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below, +for class II. If + +$$ + H = \langle u,v \rangle \cong 3^2:2S_4, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 220. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# 3^2:2S4 (class ii), order 432, expected index 220. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max7W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "BaBabab", + "babababaBaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=350)) + +actual_index = tc.number_of_classes() +print(f"The index of 3^2:2S4 (class ii) is {actual_index}") +if actual_index != 220: + raise RuntimeError(f"expected index 220, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 19 | 0 | 19 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 20 | 56 | 26.3% + #0: ToddCoxeter: time | run 0 = 27µs | all runs = 27µs | elapsed = 79µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 880 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.100 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +860 | -56 | +73.7% + #1: ToddCoxeter: phase 0.1 = 45ms | run 0 = 45ms | all runs = 45ms | elapsed = 45ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 42ms (92%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 45ms | run 0 = 45ms | all runs = 45ms | elapsed = 45ms + The index of 3^2:2S4 (class ii) is 220 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $220$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/396.md b/docs/examples/atlas/mathieu/m12/396.md new file mode 100644 index 0000000..c954b1d --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/396.md @@ -0,0 +1,135 @@ +# $2 \times S_5$ as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max8W1). + +## The computation + +The recorded run took about 69 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong 2 \times S_5, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 396. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# 2 x S5, order 240, expected index 396. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max8W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "abaBababaBababaBab", + "babaBabaBabaBaBaBaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=480)) + +actual_index = tc.number_of_classes() +print(f"The index of 2 x S5 is {actual_index}") +if actual_index != 396: + raise RuntimeError(f"expected index 396, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 38 | 0 | 38 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 39 | 113 | 25.7% + #0: ToddCoxeter: time | run 0 = 27µs | all runs = 27µs | elapsed = 80µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,584 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.111 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +1,545 | -113 | +74.3% + #1: ToddCoxeter: phase 0.1 = 68ms | run 0 = 68ms | all runs = 68ms | elapsed = 68ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 65ms (95%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 69ms | run 0 = 69ms | all runs = 69ms | elapsed = 69ms + The index of 2 x S5 is 396 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $396$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/495-i.md b/docs/examples/atlas/mathieu/m12/495-i.md new file mode 100644 index 0000000..329faad --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/495-i.md @@ -0,0 +1,139 @@ +# $2^{1+4}:S_3$ as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max9W2). + +## The computation + +The recorded run took about 96 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong 2^{1+4}:S_3, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 495. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# 2^(1+4):S3, order 192, expected index 495. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max9W2 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "abaBabaBaBabab", + ( + "BabaBaBaBabaBaBaBabaBaBaBabaBaBaBabaBaBaBabaBaBaBa" + "baBaBaBaBaBaBabaBababaBabababaBabababaBabababaBaba" + "babaBabababaBabababaBabababaBab" + ), +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=700)) + +actual_index = tc.number_of_classes() +print(f"The index of 2^(1+4):S3 is {actual_index}") +if actual_index != 495: + raise RuntimeError(f"expected index 495, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 144 | 0 | 144 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 145 | 431 | 25.2% + #0: ToddCoxeter: time | run 0 = 30µs | all runs = 30µs | elapsed = 85µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,980 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.108 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +1,835 | -431 | +74.8% + #1: ToddCoxeter: phase 0.1 = 95ms | run 0 = 95ms | all runs = 95ms | elapsed = 95ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 90ms (93%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 96ms | run 0 = 96ms | all runs = 96ms | elapsed = 96ms + The index of 2^(1+4):S3 is 495 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $495$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/495-ii.md b/docs/examples/atlas/mathieu/m12/495-ii.md new file mode 100644 index 0000000..ae4099c --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/495-ii.md @@ -0,0 +1,135 @@ +# $4^2:D_{12}$ as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max10W1). + +## The computation + +The recorded run took about 27 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong 4^2:D_{12}, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 495. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# 4^2:D12, order 192, expected index 495. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max10W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "abaBababaBababaBab", + "bababaBabaBaBaBaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=200)) + +actual_index = tc.number_of_classes() +print(f"The index of 4^2:D12 is {actual_index}") +if actual_index != 495: + raise RuntimeError(f"expected index 495, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 36 | 0 | 36 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 37 | 107 | 25.7% + #0: ToddCoxeter: time | run 0 = 26µs | all runs = 26µs | elapsed = 77µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,980 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.103 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +1,943 | -107 | +74.3% + #1: ToddCoxeter: phase 0.1 = 27ms | run 0 = 27ms | all runs = 27ms | elapsed = 27ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 25ms (93%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 27ms | run 0 = 27ms | all runs = 27ms | elapsed = 27ms + The index of 4^2:D12 is 495 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $495$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/66-i.md b/docs/examples/atlas/mathieu/m12/66-i.md new file mode 100644 index 0000000..bfe4b27 --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/66-i.md @@ -0,0 +1,136 @@ +# $A_6.2^2$ (class I) as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max3W1). + +## The computation + +The recorded run took about 1 millisecond. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below, +for class I. If + +$$ + H = \langle u,v \rangle \cong A_6.2^2, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 66. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# A6.2^2 (class i), order 1440, expected index 66. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max3W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "BaBabab", + "BababaBaBabaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=4)) + +actual_index = tc.number_of_classes() +print(f"The index of A6.2^2 (class i) is {actual_index}") +if actual_index != 66: + raise RuntimeError(f"expected index 66, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 19 | 0 | 19 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 20 | 56 | 26.3% + #0: ToddCoxeter: time | run 0 = 27µs | all runs = 27µs | elapsed = 83µs + #1: ToddCoxeter: HLT 0.0.0 | active | killed | defined + [... lines omitted ...] + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 1ms (88%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 1ms | run 0 = 1ms | all runs = 1ms | elapsed = 1ms + #1: ToddCoxeter: HLT 0.1.61 | active | killed | defined + #1: ToddCoxeter: nodes | 66 | 48,935 | 49,001 + #1: ToddCoxeter: diff 0.1.60 | +0 | +0 | +0 + #1: ToddCoxeter: diff 0.1.0 | +47 | +48,935 | +48,982 + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 264 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.60 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +244 | -56 | +73.7% + #1: ToddCoxeter: phase 0.1 = 1ms | run 0 = 1ms | all runs = 1ms | elapsed = 1ms + The index of A6.2^2 (class i) is 66 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $66$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/66-ii.md b/docs/examples/atlas/mathieu/m12/66-ii.md new file mode 100644 index 0000000..bed1c5a --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/66-ii.md @@ -0,0 +1,136 @@ +# $A_6.2^2$ (class II) as a subgroup of Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max4W1). + +## The computation + +The recorded run took about 78 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below, +for class II. If + +$$ + H = \langle u,v \rangle \cong A_6.2^2, +$$ + +then + +$$ + [\mathrm{M}_{12}:H] = 66. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +# A6.2^2 (class ii), order 1440, expected index 66. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M12/words/M12G1-max4W1 +# ATLAS output words simplified using a^2 = b^3 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "BaBaBababab", + "baBabaBabaBaBaBaB", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(microseconds=550)) + +actual_index = tc.number_of_classes() +print(f"The index of A6.2^2 (class ii) is {actual_index}") +if actual_index != 66: + raise RuntimeError(f"expected index 66, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 27 | 0 | 27 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 28 | 80 | 25.9% + #0: ToddCoxeter: time | run 0 = 27µs | all runs = 27µs | elapsed = 76µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 264 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.112 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +236 | -80 | +74.1% + #1: ToddCoxeter: phase 0.1 = 78ms | run 0 = 78ms | all runs = 78ms | elapsed = 78ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 74ms (95%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 78ms | run 0 = 78ms | all runs = 78ms | elapsed = 78ms + The index of A6.2^2 (class ii) is 66 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $66$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/95040.md b/docs/examples/atlas/mathieu/m12/95040.md new file mode 100644 index 0000000..30d14ab --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/95040.md @@ -0,0 +1,120 @@ +# The trivial subgroup of the Mathieu group M~12~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). +Enumerating over the trivial subgroup computes the order of the group. + +## The computation + +The recorded run took about 581 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $H$ be the trivial subgroup. We verify that the +[presentation of M~12~](index.md#presentation) defines a group of order + +$$ + [\mathrm{M}_{12}:H] = |\mathrm{M}_{12}| = 95,040. +$$ + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^3"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ABab)^6"), "") +presentation.add_rule(p, parse_relations("(ababaB)^6"), "") + +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.report_every(timedelta(milliseconds=5)) +tc.run_for(timedelta(seconds=10)) + +if not tc.finished(): + raise RuntimeError("The enumeration did not finish within 10 seconds") + +actual_order = tc.number_of_classes() +print(f"The size of the group defined by the presentation is {actual_order}") +if actual_order != 95040: + raise RuntimeError(f"expected order 95040, got {actual_order}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: running for approx. 10.000s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 9, |u| + |v| ∈ [2, 36], ∑(|u| + |v|) = 95 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 4 | 0.0% + #0: ToddCoxeter: time | run 0 = 22µs | all runs = 22µs | elapsed = 77µs + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 380,160 | 0 | 100.0% + #1: ToddCoxeter: diff 0.3.8 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.3.0 | -17 | -7 | +0.0% + #1: ToddCoxeter: phase 0.3 = 38ms | run 0 = 580ms | all runs = 580ms | elapsed = 580ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 2 | 0 + #0: ToddCoxeter: time spent in phases | 343ms (59%) | - (0%) | 207ms (36%) | - (0%) + #0: ToddCoxeter: phase 0.3 = 40ms | run 0 = 581ms | all runs = 581ms | elapsed = 581ms + The size of the group defined by the presentation is 95040 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed order is the claimed order: $95,040$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m12/index.md b/docs/examples/atlas/mathieu/m12/index.md new file mode 100644 index 0000000..1bf3445 --- /dev/null +++ b/docs/examples/atlas/mathieu/m12/index.md @@ -0,0 +1,48 @@ +# Mathieu group M~12~ + + + +The presentation and maximal subgroup data come from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). + +## Presentation + +$$ + \mathrm{M}_{12} = \langle a,b \mid a^2 = b^3 = (ab)^{11} = [a,b]^6 = (ababab^{-1})^6 = 1 \rangle +$$ + +with $|\mathrm{M}_{12}| = 95,040$. + +## Subgroup enumerations + +The following table summarises the enumeration over the trivial subgroup and +the eleven maximal-subgroup enumerations. Where two classes have the same +subgroup name and index, they are listed separately. Each linked page includes +self-contained code, recorded output, and an animation of the corresponding +computation. + +
+ +| Subgroup | Subgroup order | Index in $\mathrm{M}_{12}$ | Verification | +| :------- | -------------: | ------------------------: | :----------: | +| [Trivial](95040.md) | $1$ | $95,040$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$\mathrm{M}_{11}$ (class I)](12-i.md) | $7,920$ | $12$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$\mathrm{M}_{11}$ (class II)](12-ii.md) | $7,920$ | $12$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$A_6.2^2$ (class I)](66-i.md) | $1,440$ | $66$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$A_6.2^2$ (class II)](66-ii.md) | $1,440$ | $66$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$L_2(11)$](144.md) | $660$ | $144$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$3^2:2S_4$ (class I)](220-i.md) | $432$ | $220$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$3^2:2S_4$ (class II)](220-ii.md) | $432$ | $220$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$2 \times S_5$](396.md) | $240$ | $396$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$2^{1+4}:S_3$](495-i.md) | $192$ | $495$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$4^2:D_{12}$](495-ii.md) | $192$ | $495$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$A_4 \times S_3$](1320.md) | $72$ | $1,320$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | + +**Key** + +| Symbol | Meaning | +| :----: | :------ | +| :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | We could verify the index. | + +
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f389e8e5ab1ab59e8d1b3b48f9f3a0b91b9456e2 Mon Sep 17 00:00:00 2001 From: James Mitchell Date: Fri, 30 Oct 2026 19:32:18 +0000 Subject: [PATCH 3/5] Add m22 movies --- docs/.nav.yml | 12 +- docs/examples/atlas/mathieu/m22.md | 583 ------------------ docs/examples/atlas/mathieu/m22/176-i.md | 138 +++++ docs/examples/atlas/mathieu/m22/176-ii.md | 138 +++++ docs/examples/atlas/mathieu/m22/22.md | 137 ++++ docs/examples/atlas/mathieu/m22/231.md | 137 ++++ docs/examples/atlas/mathieu/m22/330.md | 137 ++++ docs/examples/atlas/mathieu/m22/443520.md | 122 ++++ docs/examples/atlas/mathieu/m22/616.md | 137 ++++ docs/examples/atlas/mathieu/m22/672.md | 137 ++++ docs/examples/atlas/mathieu/m22/77.md | 137 ++++ docs/examples/atlas/mathieu/m22/index.md | 44 ++ docs/examples/atlas/mathieu/m22/m22-176-i.mp4 | Bin 0 -> 517998 bytes .../examples/atlas/mathieu/m22/m22-176-ii.mp4 | Bin 0 -> 314019 bytes docs/examples/atlas/mathieu/m22/m22-22.mp4 | Bin 0 -> 342981 bytes docs/examples/atlas/mathieu/m22/m22-231.mp4 | Bin 0 -> 247068 bytes docs/examples/atlas/mathieu/m22/m22-330.mp4 | Bin 0 -> 503967 bytes docs/examples/atlas/mathieu/m22/m22-616.mp4 | Bin 0 -> 236034 bytes docs/examples/atlas/mathieu/m22/m22-672.mp4 | Bin 0 -> 318114 bytes docs/examples/atlas/mathieu/m22/m22-77.mp4 | Bin 0 -> 594146 bytes docs/examples/atlas/mathieu/m22/m22.mp4 | Bin 0 -> 855371 bytes 21 files changed, 1275 insertions(+), 584 deletions(-) delete mode 100644 docs/examples/atlas/mathieu/m22.md create mode 100644 docs/examples/atlas/mathieu/m22/176-i.md create mode 100644 docs/examples/atlas/mathieu/m22/176-ii.md create mode 100644 docs/examples/atlas/mathieu/m22/22.md create mode 100644 docs/examples/atlas/mathieu/m22/231.md create mode 100644 docs/examples/atlas/mathieu/m22/330.md create mode 100644 docs/examples/atlas/mathieu/m22/443520.md create mode 100644 docs/examples/atlas/mathieu/m22/616.md create mode 100644 docs/examples/atlas/mathieu/m22/672.md create mode 100644 docs/examples/atlas/mathieu/m22/77.md create mode 100644 docs/examples/atlas/mathieu/m22/index.md create mode 100644 docs/examples/atlas/mathieu/m22/m22-176-i.mp4 create mode 100644 docs/examples/atlas/mathieu/m22/m22-176-ii.mp4 create mode 100644 docs/examples/atlas/mathieu/m22/m22-22.mp4 create mode 100644 docs/examples/atlas/mathieu/m22/m22-231.mp4 create mode 100644 docs/examples/atlas/mathieu/m22/m22-330.mp4 create mode 100644 docs/examples/atlas/mathieu/m22/m22-616.mp4 create mode 100644 docs/examples/atlas/mathieu/m22/m22-672.mp4 create mode 100644 docs/examples/atlas/mathieu/m22/m22-77.mp4 create mode 100644 docs/examples/atlas/mathieu/m22/m22.mp4 diff --git a/docs/.nav.yml b/docs/.nav.yml index 7282a0a..aa0bb2b 100644 --- a/docs/.nav.yml +++ b/docs/.nav.yml @@ -42,7 +42,17 @@ nav: - "2^(1+4):S3": examples/atlas/mathieu/m12/495-i.md - "4^2:D12": examples/atlas/mathieu/m12/495-ii.md - "A4 x S3": examples/atlas/mathieu/m12/1320.md - - "Mathieu group M22": examples/atlas/mathieu/m22.md + - "Mathieu group M22": + - examples/atlas/mathieu/m22/index.md + - "Trivial": examples/atlas/mathieu/m22/443520.md + - "L3(4)": examples/atlas/mathieu/m22/22.md + - "2^4:A6": examples/atlas/mathieu/m22/77.md + - "A7 (I)": examples/atlas/mathieu/m22/176-i.md + - "A7 (II)": examples/atlas/mathieu/m22/176-ii.md + - "2^4:S5": examples/atlas/mathieu/m22/231.md + - "2^3:L3(2)": examples/atlas/mathieu/m22/330.md + - "M10 = A6.2_3": examples/atlas/mathieu/m22/616.md + - "L2(11)": examples/atlas/mathieu/m22/672.md - "Mathieu group M23": - examples/atlas/mathieu/m23/index.md - "Presentation 1": examples/atlas/mathieu/m23/p1.md diff --git a/docs/examples/atlas/mathieu/m22.md b/docs/examples/atlas/mathieu/m22.md deleted file mode 100644 index e9996da..0000000 --- a/docs/examples/atlas/mathieu/m22.md +++ /dev/null @@ -1,583 +0,0 @@ -# Mathieu group M~22~ - -The claims on this page come from -[https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). - -## Claim - -$$ - \mathrm{M}_{22} = \langle a, b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^5 = [a,bab]^3 = (ababab^{−1})^5 = 1 \rangle -$$ - -with - -$$ - |\mathrm{M}_{22}| = 443,520. -$$ - -On this page, we verify that the above claimed presentation of the Mathieu group -M~22~ defines a group of order $443,520$. - -## The code - -In [libsemigroups_pybind11][], the following script constructs the presentation -for M~11~ and runs the Todd-Coxeter algorithm. - -

-Code - -```python -from libsemigroups_pybind11 import ( - Presentation, - ToddCoxeter, - congruence_kind, - presentation, -) -from libsemigroups_pybind11.words import parse_relations - -# Setup the presentation object with the empty word and inverses, so it can represent a group -p = Presentation("abAB") -p.contains_empty_word(True) -presentation.add_inverse_rules(p, "ABab") - -# Add the defining relations -presentation.add_rule(p, parse_relations("a^2"), "") -presentation.add_rule(p, parse_relations("b^4"), "") -presentation.add_rule(p, parse_relations("(ab)^11"), "") -presentation.add_rule(p, parse_relations("(ab^2)^5"), "") -presentation.add_rule(p, parse_relations("(a,bab)^3"), "") -presentation.add_rule(p, parse_relations("(ababaB)^5"), "") - - -# Run the Todd-Coxeter algorithm -tc = ToddCoxeter(congruence_kind.twosided, p) -tc.run() - -print(f"The size of the group is {tc.number_of_classes()}") -``` - -
- -## The output - -Running the above script produces the following output: - -??? info "Output from the Python script" - ``` - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: running for approx. 30.000s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 START (strategy() = hlt) - #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 START - #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined - #0: ToddCoxeter: nodes | 1 | 0 | 1 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 0 | 4 | 0.0% - #0: ToddCoxeter: time | run 0 = 19µs | all runs = 19µs | elapsed = 89µs - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.0 STOP - #0: ToddCoxeter: HLT 0.0.1 | active | killed | defined - #0: ToddCoxeter: nodes | 5,000,018 | 1,632,158 | 6,632,176 - #0: ToddCoxeter: diff 0.0.0 | +5,000,017 | +1,632,158 | +6,632,175 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 5,679,076 | 14,320,996 | 28.4% - #0: ToddCoxeter: diff 0.0.0 | +5,679,076 | +14,320,992 | +28.4% - #0: ToddCoxeter: phase 0.0 = 547ms | run 0 = 547ms | all runs = 547ms | elapsed = 547ms - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined - #0: ToddCoxeter: nodes | 5,000,018 | 1,632,158 | 6,632,176 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 5,679,076 | 14,320,996 | 28.4% - #0: ToddCoxeter: time | run 0 = 547ms | all runs = 547ms | elapsed = 547ms - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 5,000,018 - #0: ToddCoxeter: n = lookahead_next() = 5,000,000 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined - #1: ToddCoxeter: nodes | 2,644,921 | 3,987,547 | 6,632,176 - #1: ToddCoxeter: diff 0.1.0 | -2,355,097 | +2,355,389 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 7,553,691 | 3,025,993 | 71.4% - #1: ToddCoxeter: diff 0.1.0 | +1,874,615 | -11,295,003 | +43.0% - #1: ToddCoxeter: phase 0.1 = 452ms | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.001s - #1: ToddCoxeter: lookahead progress | ~60.5% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.1 STOP - #0: ToddCoxeter: LOOKAHEAD 0.1.2 | active | killed | defined - #0: ToddCoxeter: nodes | 2,423,713 | 4,208,463 | 6,632,176 - #0: ToddCoxeter: diff 0.1.1 | -221,208 | +220,916 | +0 - #0: ToddCoxeter: diff 0.1.0 | -2,576,305 | +2,576,305 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 7,815,443 | 1,879,409 | 80.6% - #0: ToddCoxeter: diff 0.1.1 | +261,752 | -1,146,584 | +9.2% - #0: ToddCoxeter: diff 0.1.0 | +2,136,367 | -12,441,587 | +52.2% - #0: ToddCoxeter: phase 0.1 = 500ms | run 0 = 1.049s | all runs = 1.049s | elapsed = 1.049s - #0: ToddCoxeter: lookahead_next() is now max(f x a = 4,847,426, m = 10,000) (-152,574) - #0: ToddCoxeter: because f x a < n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 2,423,713 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: m = lookahead_min() = 10,000 - #0: ToddCoxeter: n = lookahead_next() = 5,000,000 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.2 START - #0: ToddCoxeter: HLT 0.2.0 | active | killed | defined - #0: ToddCoxeter: nodes | 2,423,713 | 4,208,463 | 6,632,176 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 7,815,443 | 1,879,409 | 80.6% - #0: ToddCoxeter: time | run 0 = 1.049s | all runs = 1.049s | elapsed = 1.049s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.2 STOP - #0: ToddCoxeter: HLT 0.2.1 | active | killed | defined - #0: ToddCoxeter: nodes | 4,847,474 | 4,320,458 | 9,167,932 - #0: ToddCoxeter: diff 0.2.0 | +2,423,761 | +111,995 | +2,535,756 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 10,422,837 | 8,967,059 | 53.8% - #0: ToddCoxeter: diff 0.2.0 | +2,607,394 | +7,087,650 | -26.9% - #0: ToddCoxeter: phase 0.2 = 89ms | run 0 = 1.138s | all runs = 1.138s | elapsed = 1.138s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.3 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.3.0 | active | killed | defined - #0: ToddCoxeter: nodes | 4,847,474 | 4,320,458 | 9,167,932 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 10,422,837 | 8,967,059 | 53.8% - #0: ToddCoxeter: time | run 0 = 1.138s | all runs = 1.138s | elapsed = 1.138s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,847,474 - #0: ToddCoxeter: n = lookahead_next() = 4,847,426 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.3 STOP - #0: ToddCoxeter: LOOKAHEAD 0.3.1 | active | killed | defined - #0: ToddCoxeter: nodes | 3,512,400 | 5,655,532 | 9,167,932 - #0: ToddCoxeter: diff 0.3.0 | -1,335,074 | +1,335,074 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 11,491,787 | 2,557,813 | 81.8% - #0: ToddCoxeter: diff 0.3.0 | +1,068,950 | -6,409,246 | +28.0% - #0: ToddCoxeter: phase 0.3 = 718ms | run 0 = 1.856s | all runs = 1.856s | elapsed = 1.857s - #0: ToddCoxeter: lookahead_next() is now 4,847,426 (+0) - #0: ToddCoxeter: because: - #0: ToddCoxeter: 1. n <= f x a = 7,024,800 - #0: ToddCoxeter: 2. a <= n - #0: ToddCoxeter: 3. l >= (l + a) / t = 1,211,868 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 3,512,400 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 1,335,074 - #0: ToddCoxeter: n = lookahead_next() = 4,847,426 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.4 START - #0: ToddCoxeter: HLT 0.4.0 | active | killed | defined - #0: ToddCoxeter: nodes | 3,512,400 | 5,655,532 | 9,167,932 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 11,491,787 | 2,557,813 | 81.8% - #0: ToddCoxeter: time | run 0 = 1.857s | all runs = 1.857s | elapsed = 1.857s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.4 STOP - #0: ToddCoxeter: HLT 0.4.1 | active | killed | defined - #0: ToddCoxeter: nodes | 4,847,449 | 5,703,629 | 10,551,078 - #0: ToddCoxeter: diff 0.4.0 | +1,335,049 | +48,097 | +1,383,146 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 12,934,269 | 6,455,527 | 66.7% - #0: ToddCoxeter: diff 0.4.0 | +1,442,482 | +3,897,714 | -15.1% - #0: ToddCoxeter: phase 0.4 = 67ms | run 0 = 1.924s | all runs = 1.924s | elapsed = 1.924s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.5 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.5.0 | active | killed | defined - #0: ToddCoxeter: nodes | 4,847,449 | 5,703,629 | 10,551,078 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 12,934,269 | 6,455,527 | 66.7% - #0: ToddCoxeter: time | run 0 = 1.924s | all runs = 1.924s | elapsed = 1.924s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,847,449 - #0: ToddCoxeter: n = lookahead_next() = 4,847,426 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.5.1 | active | killed | defined - #1: ToddCoxeter: nodes | 4,820,487 | 5,730,591 | 10,551,078 - #1: ToddCoxeter: diff 0.5.0 | -26,962 | +26,962 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 12,908,450 | 6,373,498 | 66.9% - #1: ToddCoxeter: diff 0.5.0 | -25,819 | -82,029 | +0.2% - #1: ToddCoxeter: phase 0.5 = 77ms | run 0 = 2.001s | all runs = 2.001s | elapsed = 2.001s - #1: ToddCoxeter: lookahead progress | ~11.0% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.5 STOP - #0: ToddCoxeter: LOOKAHEAD 0.5.2 | active | killed | defined - #0: ToddCoxeter: nodes | 4,134,106 | 6,416,972 | 10,551,078 - #0: ToddCoxeter: diff 0.5.1 | -686,381 | +686,381 | +0 - #0: ToddCoxeter: diff 0.5.0 | -713,343 | +713,343 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 13,557,552 | 2,978,872 | 82.0% - #0: ToddCoxeter: diff 0.5.1 | +649,102 | -3,394,626 | +15.0% - #0: ToddCoxeter: diff 0.5.0 | +623,283 | -3,476,655 | +15.3% - #0: ToddCoxeter: phase 0.5 = 856ms | run 0 = 2.780s | all runs = 2.780s | elapsed = 2.780s - #0: ToddCoxeter: lookahead_next() is now n x f = 9,694,852 (+4,847,426) - #0: ToddCoxeter: because: l < (l + a) / t = 1,211,862 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 4,134,106 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 713,343 - #0: ToddCoxeter: n = lookahead_next() = 4,847,426 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.6 START - #0: ToddCoxeter: HLT 0.6.0 | active | killed | defined - #0: ToddCoxeter: nodes | 4,134,106 | 6,416,972 | 10,551,078 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 13,557,552 | 2,978,872 | 82.0% - #0: ToddCoxeter: time | run 0 = 2.780s | all runs = 2.780s | elapsed = 2.780s - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: HLT 0.6.1 | active | killed | defined - #1: ToddCoxeter: nodes | 8,388,608 | 6,594,998 | 14,983,606 - #1: ToddCoxeter: diff 0.6.0 | +4,254,502 | +178,026 | +4,432,528 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 18,160,579 | 15,393,853 | 54.1% - #1: ToddCoxeter: diff 0.6.0 | +4,603,027 | +12,414,981 | -27.9% - #1: ToddCoxeter: phase 0.6 = 221ms | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.6 STOP - #0: ToddCoxeter: HLT 0.6.2 | active | killed | defined - #0: ToddCoxeter: nodes | 9,694,873 | 6,657,105 | 16,351,978 - #0: ToddCoxeter: diff 0.6.1 | +1,306,265 | +62,107 | +1,368,372 - #0: ToddCoxeter: diff 0.6.0 | +5,560,767 | +240,133 | +5,800,900 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 19,575,370 | 19,204,122 | 50.5% - #0: ToddCoxeter: diff 0.6.1 | +1,414,791 | +3,810,269 | -3.6% - #0: ToddCoxeter: diff 0.6.0 | +6,017,818 | +16,225,250 | -31.5% - #0: ToddCoxeter: phase 0.6 = 520ms | run 0 = 3.301s | all runs = 3.301s | elapsed = 3.301s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.7 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.7.0 | active | killed | defined - #0: ToddCoxeter: nodes | 9,694,873 | 6,657,105 | 16,351,978 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 19,575,370 | 19,204,122 | 50.5% - #0: ToddCoxeter: time | run 0 = 3.301s | all runs = 3.301s | elapsed = 3.301s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 9,694,873 - #0: ToddCoxeter: n = lookahead_next() = 9,694,852 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.7.1 | active | killed | defined - #1: ToddCoxeter: nodes | 9,431,978 | 6,920,002 | 16,351,978 - #1: ToddCoxeter: diff 0.7.0 | -262,895 | +262,897 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 19,360,399 | 18,367,513 | 51.3% - #1: ToddCoxeter: diff 0.7.0 | -214,971 | -836,609 | +0.8% - #1: ToddCoxeter: phase 0.7 = 700ms | run 0 = 4.001s | all runs = 4.001s | elapsed = 4.001s - #1: ToddCoxeter: lookahead progress | ~41.5% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.7 STOP - #0: ToddCoxeter: LOOKAHEAD 0.7.2 | active | killed | defined - #0: ToddCoxeter: nodes | 6,784,030 | 9,567,948 | 16,351,978 - #0: ToddCoxeter: diff 0.7.1 | -2,647,948 | +2,647,946 | +0 - #0: ToddCoxeter: diff 0.7.0 | -2,910,843 | +2,910,843 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 22,154,799 | 4,981,321 | 81.6% - #0: ToddCoxeter: diff 0.7.1 | +2,794,400 | -13,386,192 | +30.3% - #0: ToddCoxeter: diff 0.7.0 | +2,579,429 | -14,222,801 | +31.2% - #0: ToddCoxeter: phase 0.7 = 1.445s | run 0 = 4.746s | all runs = 4.746s | elapsed = 4.746s - #0: ToddCoxeter: lookahead_next() is now 9,694,852 (+0) - #0: ToddCoxeter: because: - #0: ToddCoxeter: 1. n <= f x a = 13,568,060 - #0: ToddCoxeter: 2. a <= n - #0: ToddCoxeter: 3. l >= (l + a) / t = 2,423,718 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 6,784,030 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 2,910,843 - #0: ToddCoxeter: n = lookahead_next() = 9,694,852 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.8 START - #0: ToddCoxeter: HLT 0.8.0 | active | killed | defined - #0: ToddCoxeter: nodes | 6,784,030 | 9,567,948 | 16,351,978 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 22,154,799 | 4,981,321 | 81.6% - #0: ToddCoxeter: time | run 0 = 4.746s | all runs = 4.746s | elapsed = 4.746s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.8 STOP - #0: ToddCoxeter: HLT 0.8.1 | active | killed | defined - #0: ToddCoxeter: nodes | 9,694,854 | 9,675,719 | 19,370,573 - #0: ToddCoxeter: diff 0.8.0 | +2,910,824 | +107,771 | +3,018,595 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 25,297,258 | 13,482,158 | 65.2% - #0: ToddCoxeter: diff 0.8.0 | +3,142,459 | +8,500,837 | -16.4% - #0: ToddCoxeter: phase 0.8 = 130ms | run 0 = 4.876s | all runs = 4.876s | elapsed = 4.876s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.9 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.9.0 | active | killed | defined - #0: ToddCoxeter: nodes | 9,694,854 | 9,675,719 | 19,370,573 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 25,297,258 | 13,482,158 | 65.2% - #0: ToddCoxeter: time | run 0 = 4.876s | all runs = 4.876s | elapsed = 4.876s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 9,694,854 - #0: ToddCoxeter: n = lookahead_next() = 9,694,852 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.9.1 | active | killed | defined - #1: ToddCoxeter: nodes | 9,648,811 | 9,721,762 | 19,370,573 - #1: ToddCoxeter: diff 0.9.0 | -46,043 | +46,043 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 25,256,260 | 13,338,984 | 65.4% - #1: ToddCoxeter: diff 0.9.0 | -40,998 | -143,174 | +0.2% - #1: ToddCoxeter: phase 0.9 = 125ms | run 0 = 5.001s | all runs = 5.001s | elapsed = 5.002s - #1: ToddCoxeter: lookahead progress | ~9.5% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.9.2 | active | killed | defined - #1: ToddCoxeter: nodes | 9,403,427 | 9,967,160 | 19,370,573 - #1: ToddCoxeter: diff 0.9.1 | -245,384 | +245,398 | +0 - #1: ToddCoxeter: diff 0.9.0 | -291,427 | +291,441 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 25,010,253 | 12,603,455 | 66.5% - #1: ToddCoxeter: diff 0.9.1 | -246,007 | -735,529 | +1.1% - #1: ToddCoxeter: diff 0.9.0 | -287,005 | -878,703 | +1.3% - #1: ToddCoxeter: phase 0.9 = 1.126s | run 0 = 6.002s | all runs = 6.002s | elapsed = 6.002s - #1: ToddCoxeter: lookahead progress | ~63.4% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.9 STOP - #0: ToddCoxeter: LOOKAHEAD 0.9.3 | active | killed | defined - #0: ToddCoxeter: nodes | 8,086,523 | 11,284,050 | 19,370,573 - #0: ToddCoxeter: diff 0.9.2 | -1,316,904 | +1,316,890 | +0 - #0: ToddCoxeter: diff 0.9.0 | -1,608,331 | +1,608,331 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 26,555,088 | 5,791,004 | 82.1% - #0: ToddCoxeter: diff 0.9.2 | +1,544,835 | -6,812,451 | +15.6% - #0: ToddCoxeter: diff 0.9.0 | +1,257,830 | -7,691,154 | +16.9% - #0: ToddCoxeter: phase 0.9 = 1.690s | run 0 = 6.566s | all runs = 6.566s | elapsed = 6.566s - #0: ToddCoxeter: lookahead_next() is now n x f = 19,389,704 (+9,694,852) - #0: ToddCoxeter: because: l < (l + a) / t = 2,423,713 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 8,086,523 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 1,608,331 - #0: ToddCoxeter: n = lookahead_next() = 9,694,852 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.10 START - #0: ToddCoxeter: HLT 0.10.0 | active | killed | defined - #0: ToddCoxeter: nodes | 8,086,523 | 11,284,050 | 19,370,573 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 26,555,088 | 5,791,004 | 82.1% - #0: ToddCoxeter: time | run 0 = 6.566s | all runs = 6.566s | elapsed = 6.566s - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: HLT 0.10.1 | active | killed | defined - #1: ToddCoxeter: nodes | 16,777,216 | 11,642,699 | 28,419,915 - #1: ToddCoxeter: diff 0.10.0 | +8,690,693 | +358,649 | +9,049,342 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 35,954,563 | 31,154,301 | 53.6% - #1: ToddCoxeter: diff 0.10.0 | +9,399,475 | +25,363,297 | -28.5% - #1: ToddCoxeter: phase 0.10 = 436ms | run 0 = 7.002s | all runs = 7.002s | elapsed = 7.002s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.10 STOP - #0: ToddCoxeter: HLT 0.10.2 | active | killed | defined - #0: ToddCoxeter: nodes | 19,389,750 | 11,773,835 | 31,163,585 - #0: ToddCoxeter: diff 0.10.1 | +2,612,534 | +131,136 | +2,743,670 - #0: ToddCoxeter: diff 0.10.0 | +11,303,227 | +489,785 | +11,793,012 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 38,786,740 | 38,772,260 | 50.0% - #0: ToddCoxeter: diff 0.10.1 | +2,832,177 | +7,617,959 | -3.6% - #0: ToddCoxeter: diff 0.10.0 | +12,231,652 | +32,981,256 | -32.1% - #0: ToddCoxeter: phase 0.10 = 1.051s | run 0 = 7.617s | all runs = 7.617s | elapsed = 7.617s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.11 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.11.0 | active | killed | defined - #0: ToddCoxeter: nodes | 19,389,750 | 11,773,835 | 31,163,585 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 38,786,740 | 38,772,260 | 50.0% - #0: ToddCoxeter: time | run 0 = 7.617s | all runs = 7.617s | elapsed = 7.617s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 19,389,750 - #0: ToddCoxeter: n = lookahead_next() = 19,389,704 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.11.1 | active | killed | defined - #1: ToddCoxeter: nodes | 19,119,246 | 12,044,357 | 31,163,585 - #1: ToddCoxeter: diff 0.11.0 | -270,504 | +270,522 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 38,564,145 | 37,912,839 | 50.4% - #1: ToddCoxeter: diff 0.11.0 | -222,595 | -859,421 | +0.4% - #1: ToddCoxeter: phase 0.11 = 385ms | run 0 = 8.002s | all runs = 8.002s | elapsed = 8.002s - #1: ToddCoxeter: lookahead progress | ~12.6% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.11.2 | active | killed | defined - #1: ToddCoxeter: nodes | 18,894,507 | 12,269,078 | 31,163,585 - #1: ToddCoxeter: diff 0.11.1 | -224,739 | +224,721 | +0 - #1: ToddCoxeter: diff 0.11.0 | -495,243 | +495,243 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 38,371,598 | 37,206,430 | 50.8% - #1: ToddCoxeter: diff 0.11.1 | -192,547 | -706,409 | +0.3% - #1: ToddCoxeter: diff 0.11.0 | -415,142 | -1,565,830 | +0.8% - #1: ToddCoxeter: phase 0.11 = 1.385s | run 0 = 9.003s | all runs = 9.003s | elapsed = 9.003s - #1: ToddCoxeter: lookahead progress | ~38.8% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.11.3 | active | killed | defined - #1: ToddCoxeter: nodes | 15,851,580 | 15,312,085 | 31,163,585 - #1: ToddCoxeter: diff 0.11.2 | -3,042,927 | +3,043,007 | +0 - #1: ToddCoxeter: diff 0.11.0 | -3,538,170 | +3,538,250 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 41,333,919 | 22,072,401 | 65.2% - #1: ToddCoxeter: diff 0.11.2 | +2,962,321 | -15,134,029 | +14.4% - #1: ToddCoxeter: diff 0.11.0 | +2,547,179 | -16,699,859 | +15.2% - #1: ToddCoxeter: phase 0.11 = 2.386s | run 0 = 10.003s | all runs = 10.003s | elapsed = 10.003s - #1: ToddCoxeter: lookahead progress | ~62.6% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.11 STOP - #0: ToddCoxeter: LOOKAHEAD 0.11.4 | active | killed | defined - #0: ToddCoxeter: nodes | 13,427,510 | 17,736,075 | 31,163,585 - #0: ToddCoxeter: diff 0.11.3 | -2,424,070 | +2,423,990 | +0 - #0: ToddCoxeter: diff 0.11.0 | -5,962,240 | +5,962,240 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 43,882,513 | 9,827,527 | 81.7% - #0: ToddCoxeter: diff 0.11.3 | +2,548,594 | -12,244,874 | +16.5% - #0: ToddCoxeter: diff 0.11.0 | +5,095,773 | -28,944,733 | +31.7% - #0: ToddCoxeter: phase 0.11 = 2.893s | run 0 = 10.511s | all runs = 10.511s | elapsed = 10.511s - #0: ToddCoxeter: lookahead_next() is now 19,389,704 (+0) - #0: ToddCoxeter: because: - #0: ToddCoxeter: 1. n <= f x a = 26,855,020 - #0: ToddCoxeter: 2. a <= n - #0: ToddCoxeter: 3. l >= (l + a) / t = 4,847,437 - #0: ToddCoxeter: where: a = number_of_nodes_active() = 13,427,510 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: l = nodes killed in lookahead = 5,962,240 - #0: ToddCoxeter: n = lookahead_next() = 19,389,704 - #0: ToddCoxeter: t = lookahead_growth_threshold() = 4 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.12 START - #0: ToddCoxeter: HLT 0.12.0 | active | killed | defined - #0: ToddCoxeter: nodes | 13,427,510 | 17,736,075 | 31,163,585 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 43,882,513 | 9,827,527 | 81.7% - #0: ToddCoxeter: time | run 0 = 10.511s | all runs = 10.511s | elapsed = 10.511s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.12 STOP - #0: ToddCoxeter: HLT 0.12.1 | active | killed | defined - #0: ToddCoxeter: nodes | 19,389,776 | 17,959,888 | 37,349,664 - #0: ToddCoxeter: diff 0.12.0 | +5,962,266 | +223,813 | +6,186,079 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 50,324,580 | 27,234,524 | 64.9% - #0: ToddCoxeter: diff 0.12.0 | +6,442,067 | +17,406,997 | -16.8% - #0: ToddCoxeter: phase 0.12 = 267ms | run 0 = 10.778s | all runs = 10.778s | elapsed = 10.779s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.13 START (lookahead_extent() = partial, lookahead_style() = hlt) - #0: ToddCoxeter: LOOKAHEAD 0.13.0 | active | killed | defined - #0: ToddCoxeter: nodes | 19,389,776 | 17,959,888 | 37,349,664 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 50,324,580 | 27,234,524 | 64.9% - #0: ToddCoxeter: time | run 0 = 10.778s | all runs = 10.778s | elapsed = 10.779s - #0: ToddCoxeter: because a >= n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 19,389,776 - #0: ToddCoxeter: n = lookahead_next() = 19,389,704 - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.1 | active | killed | defined - #1: ToddCoxeter: nodes | 19,314,200 | 18,035,464 | 37,349,664 - #1: ToddCoxeter: diff 0.13.0 | -75,576 | +75,576 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 50,245,486 | 27,011,314 | 65.0% - #1: ToddCoxeter: diff 0.13.0 | -79,094 | -223,210 | +0.2% - #1: ToddCoxeter: phase 0.13 = 224ms | run 0 = 11.003s | all runs = 11.003s | elapsed = 11.003s - #1: ToddCoxeter: lookahead progress | ~9.0% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.2 | active | killed | defined - #1: ToddCoxeter: nodes | 19,102,330 | 18,247,350 | 37,349,664 - #1: ToddCoxeter: diff 0.13.1 | -211,870 | +211,886 | +0 - #1: ToddCoxeter: diff 0.13.0 | -287,446 | +287,462 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 50,032,914 | 26,376,406 | 65.5% - #1: ToddCoxeter: diff 0.13.1 | -212,572 | -634,908 | +0.4% - #1: ToddCoxeter: diff 0.13.0 | -291,666 | -858,118 | +0.6% - #1: ToddCoxeter: phase 0.13 = 1.225s | run 0 = 12.003s | all runs = 12.003s | elapsed = 12.003s - #1: ToddCoxeter: lookahead progress | ~31.3% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.3 | active | killed | defined - #1: ToddCoxeter: nodes | 18,872,056 | 18,477,610 | 37,349,664 - #1: ToddCoxeter: diff 0.13.2 | -230,274 | +230,260 | +0 - #1: ToddCoxeter: diff 0.13.0 | -517,720 | +517,722 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 49,788,690 | 25,699,534 | 66.0% - #1: ToddCoxeter: diff 0.13.2 | -244,224 | -676,872 | +0.5% - #1: ToddCoxeter: diff 0.13.0 | -535,890 | -1,534,990 | +1.1% - #1: ToddCoxeter: phase 0.13 = 2.225s | run 0 = 13.004s | all runs = 13.004s | elapsed = 13.004s - #1: ToddCoxeter: lookahead progress | ~58.4% - #0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()! - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.4 | active | killed | defined - #1: ToddCoxeter: nodes | 12,447,893 | 24,901,915 | 37,349,664 - #1: ToddCoxeter: diff 0.13.3 | -6,424,163 | +6,424,305 | +0 - #1: ToddCoxeter: diff 0.13.0 | -6,941,883 | +6,942,027 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 0 | 49,791,572 | 0.0% - #1: ToddCoxeter: diff 0.13.3 | -49,788,690 | +24,092,038 | -66.0% - #1: ToddCoxeter: diff 0.13.0 | -50,324,580 | +22,557,048 | -64.9% - #1: ToddCoxeter: phase 0.13 = 3.225s | run 0 = 14.004s | all runs = 14.004s | elapsed = 14.004s - #1: ToddCoxeter: lookahead progress | ~66.6% - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: LOOKAHEAD 0.13.5 | active | killed | defined - #1: ToddCoxeter: nodes | 5,024,869 | 32,324,910 | 37,349,664 - #1: ToddCoxeter: diff 0.13.4 | -7,423,024 | +7,422,995 | +0 - #1: ToddCoxeter: diff 0.13.0 | -14,364,907 | +14,365,022 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 0 | 20,099,476 | 0.0% - #1: ToddCoxeter: diff 0.13.4 | +0 | -29,692,096 | +0.0% - #1: ToddCoxeter: diff 0.13.0 | -50,324,580 | -7,135,048 | -64.9% - #1: ToddCoxeter: phase 0.13 = 4.226s | run 0 = 15.004s | all runs = 15.004s | elapsed = 15.004s - #1: ToddCoxeter: lookahead progress | ~66.6% - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: LOOKAHEAD 0.13 STOP - #0: ToddCoxeter: LOOKAHEAD 0.13.6 | active | killed | defined - #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #0: ToddCoxeter: diff 0.13.5 | -4,581,349 | +4,581,234 | +0 - #0: ToddCoxeter: diff 0.13.0 | -18,946,256 | +18,946,256 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #0: ToddCoxeter: diff 0.13.5 | +1,774,080 | -20,099,476 | +100.0% - #0: ToddCoxeter: diff 0.13.0 | -48,550,500 | -27,234,524 | +35.1% - #0: ToddCoxeter: phase 0.13 = 5.080s | run 0 = 15.858s | all runs = 15.858s | elapsed = 15.858s - #0: ToddCoxeter: lookahead_next() is now max(f x a = 887,040, m = 10,000) (-18,502,664) - #0: ToddCoxeter: because f x a < n - #0: ToddCoxeter: where: a = number_of_nodes_active() = 443,520 - #0: ToddCoxeter: f = lookahead_growth_factor() = 2 - #0: ToddCoxeter: m = lookahead_min() = 10,000 - #0: ToddCoxeter: n = lookahead_next() = 19,389,704 - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.14 START - #0: ToddCoxeter: HLT 0.14.0 | active | killed | defined - #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #0: ToddCoxeter: time | run 0 = 15.858s | all runs = 15.858s | elapsed = 15.858s - ++++++++++++++++++++++++++++++++ - #1: ToddCoxeter: HLT 0.14.1 | active | killed | defined - #1: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #1: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0 - #1: ToddCoxeter: | active | missing | % complete - #1: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #1: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0.0% - #1: ToddCoxeter: phase 0.14 = 146ms | run 0 = 16.004s | all runs = 16.004s | elapsed = 16.005s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.14 STOP - #0: ToddCoxeter: HLT 0.14.2 | active | killed | defined - #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #0: ToddCoxeter: diff 0.14.1 | +0 | +0 | +0 - #0: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #0: ToddCoxeter: diff 0.14.1 | +0 | +0 | +0.0% - #0: ToddCoxeter: diff 0.14.0 | +0 | +0 | +0.0% - #0: ToddCoxeter: phase 0.14 = 151ms | run 0 = 16.009s | all runs = 16.009s | elapsed = 16.010s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: HLT 0.15.3 | active | killed | defined - #0: ToddCoxeter: nodes | 443,520 | 36,906,144 | 37,349,664 - #0: ToddCoxeter: diff 0.15.2 | +0 | +0 | +0 - #0: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0 - #0: ToddCoxeter: | active | missing | % complete - #0: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% - #0: ToddCoxeter: diff 0.15.2 | +0 | +0 | +0.0% - #0: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0.0% - #0: ToddCoxeter: phase 0.15 = 151ms | run 0 = 16.010s | all runs = 16.010s | elapsed = 16.010s - ++++++++++++++++++++++++++++++++ - #0: ToddCoxeter: RUN 0 STOP (finished) - #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch - #0: ToddCoxeter: num. phases | 7 | 0 | 8 | 0 - #0: ToddCoxeter: time spent in phases | 13.184s (82%) | - (0%) | 2.826s (18%) | - (0%) - #0: ToddCoxeter: phase 0.15 = 160ms | run 0 = 16.019s | all runs = 16.019s | elapsed = 16.019s - The size of the group is 443520 - ``` - -:material-checkbox-marked-circle-outline: The computed size of the group matches -the size of the group provided on the ATLAS: $443,520$. diff --git a/docs/examples/atlas/mathieu/m22/176-i.md b/docs/examples/atlas/mathieu/m22/176-i.md new file mode 100644 index 0000000..7475576 --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/176-i.md @@ -0,0 +1,138 @@ +# $A_7$ (class I) as a subgroup of Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max3W1). + +## The computation + +The recorded run took about 1.498 seconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below, +for class I. If + +$$ + H = \langle u,v \rangle \cong A_7, +$$ + +then + +$$ + [\mathrm{M}_{22}:H] = 176. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +# A7 (class i), order 2520, expected index 176. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max3W1 +# ATLAS output words simplified using a^2 = b^4 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "bbaBabbaBabbaBabbaBabbaBabbaBabbaBaBabababababbababbababbababbababbababbababb", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(milliseconds=15)) + +actual_index = tc.number_of_classes() +print(f"The index of A7 (class i) is {actual_index}") +if actual_index != 176: + raise RuntimeError(f"expected index 176, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 77 | 0 | 77 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 78 | 230 | 25.3% + #0: ToddCoxeter: time | run 0 = 29µs | all runs = 29µs | elapsed = 82µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 704 | 0 | 100.0% + #1: ToddCoxeter: diff 0.7.3 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.7.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.7 = 24ms | run 0 = 1.485s | all runs = 1.485s | elapsed = 1.485s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 3 | 0 | 4 | 0 + #0: ToddCoxeter: time spent in phases | 1.213s (81%) | - (0%) | 246ms (16%) | - (0%) + #0: ToddCoxeter: phase 0.7 = 37ms | run 0 = 1.498s | all runs = 1.498s | elapsed = 1.498s + The index of A7 (class i) is 176 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $176$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/176-ii.md b/docs/examples/atlas/mathieu/m22/176-ii.md new file mode 100644 index 0000000..2eea4bb --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/176-ii.md @@ -0,0 +1,138 @@ +# $A_7$ (class II) as a subgroup of Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max4W1). + +## The computation + +The recorded run took about 347 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below, +for class II. If + +$$ + H = \langle u,v \rangle \cong A_7, +$$ + +then + +$$ + [\mathrm{M}_{22}:H] = 176. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +# A7 (class ii), order 2520, expected index 176. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max4W1 +# ATLAS output words simplified using a^2 = b^4 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "bbaBaBaBababababababb", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(milliseconds=3)) + +actual_index = tc.number_of_classes() +print(f"The index of A7 (class ii) is {actual_index}") +if actual_index != 176: + raise RuntimeError(f"expected index 176, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 21 | 0 | 21 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 22 | 62 | 26.2% + #0: ToddCoxeter: time | run 0 = 26µs | all runs = 26µs | elapsed = 80µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 704 | 0 | 100.0% + #1: ToddCoxeter: diff 0.3.9 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.3.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.3 = 26ms | run 0 = 346ms | all runs = 346ms | elapsed = 346ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 2 | 0 + #0: ToddCoxeter: time spent in phases | 159ms (46%) | - (0%) | 160ms (46%) | - (0%) + #0: ToddCoxeter: phase 0.3 = 27ms | run 0 = 347ms | all runs = 347ms | elapsed = 347ms + The index of A7 (class ii) is 176 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $176$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/22.md b/docs/examples/atlas/mathieu/m22/22.md new file mode 100644 index 0000000..5e3d635 --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/22.md @@ -0,0 +1,137 @@ +# $L_3(4)$ as a subgroup of Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max1W1). + +## The computation + +The recorded run took about 509 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong L_3(4), +$$ + +then + +$$ + [\mathrm{M}_{22}:H] = 22. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +# L3(4), order 20160, expected index 22. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max1W1 +# ATLAS output words simplified using a^2 = b^4 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "bbaBabbaBaBaBaBababababababbababb", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(milliseconds=5)) + +actual_index = tc.number_of_classes() +print(f"The index of L3(4) is {actual_index}") +if actual_index != 22: + raise RuntimeError(f"expected index 22, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 33 | 0 | 33 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 34 | 98 | 25.8% + #0: ToddCoxeter: time | run 0 = 27µs | all runs = 27µs | elapsed = 80µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 88 | 0 | 100.0% + #1: ToddCoxeter: diff 0.3.6 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.3.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.3 = 27ms | run 0 = 507ms | all runs = 507ms | elapsed = 507ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 2 | 0 + #0: ToddCoxeter: time spent in phases | 333ms (65%) | - (0%) | 146ms (29%) | - (0%) + #0: ToddCoxeter: phase 0.3 = 28ms | run 0 = 508ms | all runs = 508ms | elapsed = 509ms + The index of L3(4) is 22 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $22$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/231.md b/docs/examples/atlas/mathieu/m22/231.md new file mode 100644 index 0000000..4cbf592 --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/231.md @@ -0,0 +1,137 @@ +# $2^4:S_5$ as a subgroup of Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max5W1). + +## The computation + +The recorded run took about 194 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong 2^4:S_5, +$$ + +then + +$$ + [\mathrm{M}_{22}:H] = 231. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +# 2^4:S5, order 1920, expected index 231. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max5W1 +# ATLAS output words simplified using a^2 = b^4 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "ababbababababbababbabbab", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(milliseconds=1.5)) + +actual_index = tc.number_of_classes() +print(f"The index of 2^4:S5 is {actual_index}") +if actual_index != 231: + raise RuntimeError(f"expected index 231, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 23 | 0 | 23 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 24 | 68 | 26.1% + #0: ToddCoxeter: time | run 0 = 27µs | all runs = 27µs | elapsed = 83µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 924 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.103 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +900 | -68 | +73.9% + #1: ToddCoxeter: phase 0.1 = 193ms | run 0 = 193ms | all runs = 193ms | elapsed = 193ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 180ms (93%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 194ms | run 0 = 194ms | all runs = 194ms | elapsed = 194ms + The index of 2^4:S5 is 231 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $231$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/330.md b/docs/examples/atlas/mathieu/m22/330.md new file mode 100644 index 0000000..44543aa --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/330.md @@ -0,0 +1,137 @@ +# $2^3:L_3(2)$ as a subgroup of Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max6W1). + +## The computation + +The recorded run took about 869 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong 2^3:L_3(2), +$$ + +then + +$$ + [\mathrm{M}_{22}:H] = 330. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +# 2^3:L3(2), order 1344, expected index 330. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max6W1 +# ATLAS output words simplified using a^2 = b^4 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "bbaBabbaBabbaBaBaBabababababbababbababb", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(milliseconds=8)) + +actual_index = tc.number_of_classes() +print(f"The index of 2^3:L3(2) is {actual_index}") +if actual_index != 330: + raise RuntimeError(f"expected index 330, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 39 | 0 | 39 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 40 | 116 | 25.6% + #0: ToddCoxeter: time | run 0 = 29µs | all runs = 29µs | elapsed = 81µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,320 | 0 | 100.0% + #1: ToddCoxeter: diff 0.5.5 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.5.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.5 = 31ms | run 0 = 865ms | all runs = 865ms | elapsed = 865ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 2 | 0 | 3 | 0 + #0: ToddCoxeter: time spent in phases | 627ms (72%) | - (0%) | 205ms (24%) | - (0%) + #0: ToddCoxeter: phase 0.5 = 36ms | run 0 = 869ms | all runs = 869ms | elapsed = 869ms + The index of 2^3:L3(2) is 330 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $330$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/443520.md b/docs/examples/atlas/mathieu/m22/443520.md new file mode 100644 index 0000000..52b4989 --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/443520.md @@ -0,0 +1,122 @@ +# The trivial subgroup of the Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +Enumerating over the trivial subgroup computes the order of the group. + +## The computation + +The recorded run took about 9.392 seconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $H$ be the trivial subgroup. We verify that the +[presentation of M~22~](index.md#presentation) defines a group of order + +$$ + [\mathrm{M}_{22}:H] = |\mathrm{M}_{22}| = 443,520. +$$ + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +tc = ToddCoxeter(congruence_kind.twosided, p) +tc.report_every(timedelta(milliseconds=125)) +tc.run_for(timedelta(seconds=30)) + +if not tc.finished(): + raise RuntimeError("The enumeration did not finish within 30 seconds") + +actual_order = tc.number_of_classes() +print(f"The size of the group defined by the presentation is {actual_order}") +if actual_order != 443520: + raise RuntimeError(f"expected order 443520, got {actual_order}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: running for approx. 30.000s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 1 | 0 | 1 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 0 | 4 | 0.0% + #0: ToddCoxeter: time | run 0 = 19µs | all runs = 19µs | elapsed = 78µs + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 1,774,080 | 0 | 100.0% + #1: ToddCoxeter: diff 0.15.3 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.15.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.15 = 175ms | run 0 = 9.349s | all runs = 9.349s | elapsed = 9.349s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 7 | 0 | 8 | 0 + #0: ToddCoxeter: time spent in phases | 8.202s (87%) | - (0%) | 1.055s (11%) | - (0%) + #0: ToddCoxeter: phase 0.15 = 217ms | run 0 = 9.392s | all runs = 9.392s | elapsed = 9.392s + The size of the group defined by the presentation is 443520 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed order is the claimed order: $443,520$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/616.md b/docs/examples/atlas/mathieu/m22/616.md new file mode 100644 index 0000000..d0a7a65 --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/616.md @@ -0,0 +1,137 @@ +# $\mathrm{M}_{10} = A_6.2_3$ as a subgroup of Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max7W1). + +## The computation + +The recorded run took about 78 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong \mathrm{M}_{10} = A_6.2_3, +$$ + +then + +$$ + [\mathrm{M}_{22}:H] = 616. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +# M10 = A6.2_3, order 720, expected index 616. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max7W1 +# ATLAS output words simplified using a^2 = b^4 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "BababbaBabb", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(milliseconds=0.6)) + +actual_index = tc.number_of_classes() +print(f"The index of M10 is {actual_index}") +if actual_index != 616: + raise RuntimeError(f"expected index 616, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 11 | 0 | 11 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 12 | 32 | 27.3% + #0: ToddCoxeter: time | run 0 = 26µs | all runs = 26µs | elapsed = 82µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 2,464 | 0 | 100.0% + #1: ToddCoxeter: diff 0.1.103 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.1.0 | +2,452 | -32 | +72.7% + #1: ToddCoxeter: phase 0.1 = 78ms | run 0 = 78ms | all runs = 78ms | elapsed = 78ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 0 | 0 | 1 | 0 + #0: ToddCoxeter: time spent in phases | - (0%) | - (0%) | 71ms (91%) | - (0%) + #0: ToddCoxeter: phase 0.1 = 78ms | run 0 = 78ms | all runs = 78ms | elapsed = 78ms + The index of M10 is 616 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $616$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/672.md b/docs/examples/atlas/mathieu/m22/672.md new file mode 100644 index 0000000..9198fae --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/672.md @@ -0,0 +1,137 @@ +# $L_2(11)$ as a subgroup of Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max8W1). + +## The computation + +The recorded run took about 392 milliseconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong L_2(11), +$$ + +then + +$$ + [\mathrm{M}_{22}:H] = 672. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +# L2(11), order 660, expected index 672. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max8W1 +# ATLAS output words simplified using a^2 = b^4 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "bbaBabbaBabbaBaBabababababbababbababb", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(milliseconds=3.5)) + +actual_index = tc.number_of_classes() +print(f"The index of L2(11) is {actual_index}") +if actual_index != 672: + raise RuntimeError(f"expected index 672, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 37 | 0 | 37 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 38 | 110 | 25.7% + #0: ToddCoxeter: time | run 0 = 27µs | all runs = 27µs | elapsed = 81µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 2,688 | 0 | 100.0% + #1: ToddCoxeter: diff 0.3.7 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.3.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.3 = 24ms | run 0 = 388ms | all runs = 388ms | elapsed = 389ms + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 1 | 0 | 2 | 0 + #0: ToddCoxeter: time spent in phases | 214ms (55%) | - (0%) | 150ms (38%) | - (0%) + #0: ToddCoxeter: phase 0.3 = 28ms | run 0 = 392ms | all runs = 392ms | elapsed = 392ms + The index of L2(11) is 672 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $672$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/77.md b/docs/examples/atlas/mathieu/m22/77.md new file mode 100644 index 0000000..c7df5af --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/77.md @@ -0,0 +1,137 @@ +# $2^4:A_6$ as a subgroup of Mathieu group M~22~ + + + +The presentation comes from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). +The subgroup generators come from the corresponding +[ATLAS program](https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max2W1). + +## The computation + +The recorded run took about 1.728 seconds. The animation shows its +progress, with the number of active nodes and the proportion of defined edges. + +
+
+ +
+
+ Progress of the Todd–Coxeter computation. +
+
+ +## Claim + +Let $u$ and $v$ be the two subgroup words listed in the code below. If + +$$ + H = \langle u,v \rangle \cong 2^4:A_6, +$$ + +then + +$$ + [\mathrm{M}_{22}:H] = 77. +$$ + +We verify the claimed index, but not the claimed isomorphism type. In the code, +uppercase letters denote inverses, so `A` represents $a^{-1}$ and `B` +represents $b^{-1}$. + +## The code + +The following script includes the full presentation and can be run on its own. + +
+Code + +```python +from datetime import timedelta + +from libsemigroups_pybind11 import ( + Presentation, + ToddCoxeter, + congruence_kind, + presentation, +) +from libsemigroups_pybind11.words import parse_relations + +# Enable the empty word and inverses so the presentation defines a group. +p = Presentation("abAB") +p.contains_empty_word(True) +presentation.add_inverse_rules(p, "ABab") + +# Add the defining relations +presentation.add_rule(p, parse_relations("a^2"), "") +presentation.add_rule(p, parse_relations("b^4"), "") +presentation.add_rule(p, parse_relations("(ab)^11"), "") +presentation.add_rule(p, parse_relations("(ab^2)^5"), "") +# [a, bab] = a^-1 (bab)^-1 a (bab). +presentation.add_rule(p, parse_relations("(ABABabab)^3"), "") +presentation.add_rule(p, parse_relations("(ababaB)^5"), "") + +# 2^4:A6, order 5760, expected index 77. +# Source: https://brauer.maths.qmul.ac.uk/Atlas/spor/M22/words/M22G1-max2W1 +# ATLAS output words simplified using a^2 = b^4 = 1; B denotes b^-1. +tc = ToddCoxeter(congruence_kind.onesided, p) +for word in ( + "a", + "bbaBabbaBabbaBabbaBabbaBabbaBaBaBaBabababababababbababbababbababbababbababb", +): + tc.add_generating_pair(word, "") + +tc.report_every(timedelta(milliseconds=18)) + +actual_index = tc.number_of_classes() +print(f"The index of 2^4:A6 is {actual_index}") +if actual_index != 77: + raise RuntimeError(f"expected index 77, got {actual_index}") +``` + +
+ +## The output + +??? info "Truncated output from the Python script" + + ``` + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 START (strategy() = hlt) + #0: ToddCoxeter: |A| = 4, |R| = 10, |u| + |v| ∈ [2, 30], ∑(|u| + |v|) = 105 + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: HLT 0.0 START + #0: ToddCoxeter: HLT 0.0.0 | active | killed | defined + #0: ToddCoxeter: nodes | 75 | 0 | 75 + #0: ToddCoxeter: | active | missing | % complete + #0: ToddCoxeter: edges | 76 | 224 | 25.3% + #0: ToddCoxeter: time | run 0 = 28µs | all runs = 28µs | elapsed = 82µs + ++++++++++++++++++++++++++++++++ + #1: ToddCoxeter: HLT 0.0.1 | active | killed | defined + [... lines omitted ...] + #1: ToddCoxeter: | active | missing | % complete + #1: ToddCoxeter: edges | 308 | 0 | 100.0% + #1: ToddCoxeter: diff 0.7.3 | +0 | +0 | +0.0% + #1: ToddCoxeter: diff 0.7.0 | +0 | +0 | +0.0% + #1: ToddCoxeter: phase 0.7 = 34ms | run 0 = 1.726s | all runs = 1.726s | elapsed = 1.726s + ++++++++++++++++++++++++++++++++ + #0: ToddCoxeter: RUN 0 STOP (finished) + #0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch + #0: ToddCoxeter: num. phases | 3 | 0 | 4 | 0 + #0: ToddCoxeter: time spent in phases | 1.444s (84%) | - (0%) | 247ms (14%) | - (0%) + #0: ToddCoxeter: phase 0.7 = 35ms | run 0 = 1.728s | all runs = 1.728s | elapsed = 1.728s + The index of 2^4:A6 is 77 + ``` + +:material-checkbox-marked-circle-outline:{ style="color: #00c853" } +The computed index is the claimed index: $77$. + +!!! note + + The output and timings may vary between machines, but the final result + should be the same. + diff --git a/docs/examples/atlas/mathieu/m22/index.md b/docs/examples/atlas/mathieu/m22/index.md new file mode 100644 index 0000000..10aaf4c --- /dev/null +++ b/docs/examples/atlas/mathieu/m22/index.md @@ -0,0 +1,44 @@ +# Mathieu group M~22~ + + + +The presentation and maximal subgroup data come from the +[ATLAS of Group Representations](https://brauer.maths.qmul.ac.uk/Atlas/v3/scripts/group2.php?id=167&subpage=11). + +## Presentation + +$$ + \mathrm{M}_{22} = \langle a,b \mid a^2 = b^4 = (ab)^{11} = (ab^2)^5 = [a,bab]^3 = (ababab^{-1})^5 = 1 \rangle +$$ + +with $|\mathrm{M}_{22}| = 443,520$. + +## Subgroup enumerations + +The following table summarises the enumeration over the trivial subgroup and +the eight maximal-subgroup enumerations. The two classes of $A_7$ are listed +separately. Each linked page includes self-contained code, recorded output, +and an animation of the corresponding computation. + +
+ +| Subgroup | Subgroup order | Index in $\mathrm{M}_{22}$ | Verification | +| :------- | -------------: | ------------------------: | :----------: | +| [Trivial](443520.md) | $1$ | $443,520$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$L_3(4)$](22.md) | $20,160$ | $22$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$2^4:A_6$](77.md) | $5,760$ | $77$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$A_7$ (class I)](176-i.md) | $2,520$ | $176$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$A_7$ (class II)](176-ii.md) | $2,520$ | $176$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$2^4:S_5$](231.md) | $1,920$ | $231$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$2^3:L_3(2)$](330.md) | $1,344$ | $330$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$\mathrm{M}_{10} = A_6.2_3$](616.md) | $720$ | $616$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | +| [$L_2(11)$](672.md) | $660$ | $672$ | :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | + +**Key** + +| Symbol | Meaning | +| :----: | :------ | +| :material-checkbox-marked-circle-outline:{ style="color: #00c853" } | We could verify the index. | + +
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160 ------------------ docs/examples/atlas/mathieu/m12/12-i.md | 136 +++++++++++++++ docs/examples/atlas/mathieu/m12/12-ii.md | 136 +++++++++++++++ docs/examples/atlas/mathieu/m12/1320.md | 135 +++++++++++++++ docs/examples/atlas/mathieu/m12/144.md | 135 +++++++++++++++ docs/examples/atlas/mathieu/m12/220-i.md | 136 +++++++++++++++ docs/examples/atlas/mathieu/m12/220-ii.md | 136 +++++++++++++++ docs/examples/atlas/mathieu/m12/396.md | 135 +++++++++++++++ docs/examples/atlas/mathieu/m12/495-i.md | 139 +++++++++++++++ docs/examples/atlas/mathieu/m12/495-ii.md | 135 +++++++++++++++ docs/examples/atlas/mathieu/m12/66-i.md | 136 +++++++++++++++ docs/examples/atlas/mathieu/m12/66-ii.md | 136 +++++++++++++++ docs/examples/atlas/mathieu/m12/95040.md | 120 +++++++++++++ docs/examples/atlas/mathieu/m12/index.md | 48 ++++++ docs/examples/atlas/mathieu/m12/m12-12-i.mp4 | Bin 0 -> 255923 bytes docs/examples/atlas/mathieu/m12/m12-12-ii.mp4 | Bin 0 -> 272004 bytes 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docs/examples/atlas/mathieu/m12/m12-396.mp4 create mode 100644 docs/examples/atlas/mathieu/m12/m12-495-i.mp4 create mode 100644 docs/examples/atlas/mathieu/m12/m12-495-ii.mp4 create mode 100644 docs/examples/atlas/mathieu/m12/m12-66-i.mp4 create mode 100644 docs/examples/atlas/mathieu/m12/m12-66-ii.mp4 create mode 100644 docs/examples/atlas/mathieu/m12/m12.mp4 diff --git a/docs/.nav.yml b/docs/.nav.yml index 0855d47..7282a0a 100644 --- a/docs/.nav.yml +++ b/docs/.nav.yml @@ -28,7 +28,20 @@ nav: - "M9:2": examples/atlas/mathieu/m11/55.md - "S5": examples/atlas/mathieu/m11/66.md - "2S4": examples/atlas/mathieu/m11/165.md - - "Mathieu group M12": examples/atlas/mathieu/m12.md + - "Mathieu group M12": + - examples/atlas/mathieu/m12/index.md + - "Trivial": examples/atlas/mathieu/m12/95040.md + - "M11 (I)": examples/atlas/mathieu/m12/12-i.md + - "M11 (II)": examples/atlas/mathieu/m12/12-ii.md + - "A6.2^2 (I)": examples/atlas/mathieu/m12/66-i.md + - "A6.2^2 (II)": examples/atlas/mathieu/m12/66-ii.md + - "L2(11)": examples/atlas/mathieu/m12/144.md + - "3^2:2S4 (I)": examples/atlas/mathieu/m12/220-i.md + - "3^2:2S4 (II)": examples/atlas/mathieu/m12/220-ii.md + - "2 x S5": examples/atlas/mathieu/m12/396.md + - "2^(1+4):S3": examples/atlas/mathieu/m12/495-i.md + - "4^2:D12": examples/atlas/mathieu/m12/495-ii.md + - "A4 x S3": examples/atlas/mathieu/m12/1320.md - "Mathieu group M22": examples/atlas/mathieu/m22.md - "Mathieu group M23": - examples/atlas/mathieu/m23/index.md diff --git a/docs/examples/atlas/mathieu/m12.md b/docs/examples/atlas/mathieu/m12.md deleted file mode 100644 index a2b1837..0000000 --- a/docs/examples/atlas/mathieu/m12.md +++ /dev/null @@ -1,160 +0,0 @@ -# Mathieu group M~12~ - -The claims on this page come from -[https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12](https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/M12/). - -## Claim - -$$ - \mathrm{M}_{12} = \langle a, b \mid a^2 = b^3 = (ab)^{11} = [a, b]^6 = (ababab^{−1})^6 = 1 \rangle -$$ - -with - -$$ - |\mathrm{M}_{12}| = 95,040. -$$ - -On this page, we verify that the above claimed presentation of the Mathieu group -M~12~ defines a group of order $95,040$. - -## The code - -In [libsemigroups_pybind11][], the following script constructs the presentation -for M~12~ and runs the Todd-Coxeter algorithm. - -