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Enumeration videos for all Mathieu groups #15

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514f647
Final m11
james-d-mitchell Oct 30, 2026
63cd16d
Final m12
james-d-mitchell Oct 30, 2026
f389e8e
Add m22 movies
james-d-mitchell Oct 30, 2026
fd4a525
Add m23
james-d-mitchell Oct 31, 2026
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Add m24
james-d-mitchell Oct 31, 2026
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68 changes: 58 additions & 10 deletions docs/.nav.yml
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Expand Up @@ -20,18 +20,66 @@ nav:
- "Verifying presentations for sporadic simple groups":
- examples/atlas/index.md
- "Mathieu groups":
- "Mathieu group M11": examples/atlas/mathieu/m11.md
- "Mathieu group M12": examples/atlas/mathieu/m12.md
- "Mathieu group M22": examples/atlas/mathieu/m22.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/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/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
- "Presentation 2": examples/atlas/mathieu/m23/p2.md
- "M11, P1": examples/atlas/mathieu/m23/p1_m11.md
- "M22, P1": examples/atlas/mathieu/m23/p1_m22.md
- "M11, P2": examples/atlas/mathieu/m23/p2_m11.md
- "M22, P2": examples/atlas/mathieu/m23/p2_m22.md
- "Mathieu group M24": examples/atlas/mathieu/m24.md
- "Trivial (P1)": examples/atlas/mathieu/m23/p1.md
- "Trivial (P2)": examples/atlas/mathieu/m23/p2.md
- "M22 (P1)": examples/atlas/mathieu/m23/p1_m22.md
- "L3(4):2_2 (P1)": examples/atlas/mathieu/m23/253-i.md
- "2^4:A7 (P1)": examples/atlas/mathieu/m23/253-ii.md
- "A8 (P1)": examples/atlas/mathieu/m23/506.md
- "M11 (P1)": examples/atlas/mathieu/m23/p1_m11.md
- "2^4:(3 x A5):2 (P1)": examples/atlas/mathieu/m23/1771.md
- "23:11 (P1)": examples/atlas/mathieu/m23/40320.md
- "Earlier P2 subgroup runs":
- "M22 (P2)": examples/atlas/mathieu/m23/p2_m22.md
- "M11 (P2)": examples/atlas/mathieu/m23/p2_m11.md
- "Mathieu group M24":
- examples/atlas/mathieu/m24/index.md
- "Trivial": examples/atlas/mathieu/m24/244823040.md
- "Cyclic (order 23)": examples/atlas/mathieu/m24/10644480.md
- "M23": examples/atlas/mathieu/m24/24.md
- "M22:2": examples/atlas/mathieu/m24/276.md
- "2^4:A8": examples/atlas/mathieu/m24/759.md
- "M12:2": examples/atlas/mathieu/m24/1288.md
- "2^6:3.S6": examples/atlas/mathieu/m24/1771.md
- "L3(4):S3": examples/atlas/mathieu/m24/2024.md
- "2^6:(L3(2) x S3)": examples/atlas/mathieu/m24/3795.md
- "L2(23)": examples/atlas/mathieu/m24/40320.md
- "L2(7)": examples/atlas/mathieu/m24/1457280.md
- "Leech lattice groups":
- examples/atlas/leech-lattice/hs.md
- "Janko group J2, Hall-Janko group HJ":
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129 changes: 129 additions & 0 deletions docs/examples/atlas/mathieu/m11/11.md
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# $\mathrm{M}_{10} = A_6.2_3$ as a subgroup of Mathieu group M~11~

<!-- AI assistance: OpenAI Codex prepared this page from the pyatlas verification scripts and logs. -->

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.

<figure style="margin-top: 0;">
<div style="overflow: hidden;">
<video autoplay loop muted playsinline
style="display: block; width: 100%;
margin-top: -10%; margin-bottom: -5%;">
<source src="../m11-11.mp4" type="video/mp4">
</video>
</div>
<figcaption style="margin-top: 0;">
Progress of the Todd–Coxeter computation.
</figcaption>
</figure>

## 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.

<details markdown>
<summary>Code</summary>

```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}")
```

</details>

## 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.

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