Mathieu group M24
The claims on this page come from https://brauer.maths.qmul.ac.uk/Atlas/spor/M24/mag/M24G1-P1.M. This is a link to a Magma file that contains a presentation (amongst other things). This presentation differs from the presentation provided in plain-text on the ATLAS webpage due to the inclusion of an additional relation. The comments in the Magma file claim that this additional relation is redundant. We do not verify that this relation is redundant here.
Claim
\[
\mathrm{M}_{24} = \langle a, b \mid a^2 = b^3 = (ab)^{23} = [a, b]^{12} = [a, bab]^5 = (ababab^{−1})^3(abab^{−1}ab^{−1})^3 = [a,b^{-1}abab]^5 = (ab(abab^{−1})^3)^4 = 1 \rangle
\]
with
\[
[\mathrm{M}_{23} : \langle ab \rangle] = 10,644,480.
\]
The code
In libsemigroups_pybind11, the following script constructs the presentation for M24, adds the generating pairs that define the subgroup \(\langle ab \rangle\), and runs the Todd-Coxeter algorithm.
Code
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)^23"), "")
presentation.add_rule(p, parse_relations("(a,b)^12"), "")
presentation.add_rule(p, parse_relations("(a,bab)^5"), "")
presentation.add_rule(p, parse_relations("(ababaB)^3(abaBaB)^3"), "")
presentation.add_rule(p, parse_relations("(a,Babab)^5"), "") # the claimed redundant relation
presentation.add_rule(p, parse_relations("(ab(abaB)^3)^4"), "")
# Make the presentation a bit shorter
presentation.replace_word_with_new_generator(p, presentation.longest_subword_reducing_length(p))
presentation.replace_word_with_new_generator(p, presentation.longest_subword_reducing_length(p))
presentation.sort_each_rule(p)
presentation.sort_rules(p)
# Add generating pairs for the subgroup
tc = ToddCoxeter(congruence_kind.onesided, p)
tc.add_generating_pair("ab", "")
# Run the Todd-Coxeter algorithm
tc.strategy(tc.options.strategy.felsch)
tc.def_version(tc.options.def_version.two)
tc.def_policy(tc.options.def_policy.discard_all_if_no_space)
tc.use_relations_in_extra(True)
tc.def_max(10000)
tc.run()
print(f"The index of the subgroup is {tc.number_of_classes()}")
The output
The truncated output of the enumeration is below:
Truncated output from the Python script
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
#0: ToddCoxeter: |A| = 6, |R| = 14, |u| + |v| ∈ [2, 45], ∑(|u| + |v|) = 178
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 START
#0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined
#0: ToddCoxeter: nodes | 101 | 35 | 136
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 157 | 449 | 25.9%
#0: ToddCoxeter: time | run 0 = 180µs | all runs = 180µs | elapsed = 344µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 1,415,661 | 175,779 | 1,591,465
#1: ToddCoxeter: diff 0.0.0 | +1,415,560 | +175,744 | +1,591,329
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 5,654,857 | 2,839,109 | 66.6%
#1: ToddCoxeter: diff 0.0.0 | +5,654,700 | +2,838,660 | +40.7%
#1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined
#1: ToddCoxeter: nodes | 2,802,886 | 379,414 | 3,182,324
#1: ToddCoxeter: diff 0.0.1 | +1,387,225 | +203,635 | +1,590,859
#1: ToddCoxeter: diff 0.0.0 | +2,802,785 | +379,379 | +3,182,188
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 11,202,682 | 5,614,634 | 66.6%
#1: ToddCoxeter: diff 0.0.1 | +5,547,825 | +2,775,525 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +11,202,525 | +5,614,185 | +40.7%
#1: ToddCoxeter: phase 0.0 = 2.000s | run 0 = 2.000s | all runs = 2.000s | elapsed = 2.001s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined
#1: ToddCoxeter: nodes | 4,194,304 | 561,668 | 4,755,972
#1: ToddCoxeter: diff 0.0.2 | +1,391,418 | +182,254 | +1,573,648
#1: ToddCoxeter: diff 0.0.0 | +4,194,203 | +561,633 | +4,755,836
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 16,763,867 | 8,401,957 | 66.6%
#1: ToddCoxeter: diff 0.0.2 | +5,561,185 | +2,787,323 | -0.0%
#1: ToddCoxeter: diff 0.0.0 | +16,763,710 | +8,401,508 | +40.7%
#1: ToddCoxeter: phase 0.0 = 3.000s | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s
++++++++++++++++++++++++++++++++
[... lines omitted ...]
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 STOP
#0: ToddCoxeter: LOOKAHEAD 0.1.65 | active | killed | defined
#0: ToddCoxeter: nodes | 10,644,480 | 52,857,227 | 63,501,707
#0: ToddCoxeter: diff 0.1.64 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 63,866,880 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.64 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 1min4s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min37s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 21,288,960, m = 10,000) (+16,288,960)
#0: ToddCoxeter: because a > n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 10,644,480
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: m = lookahead_min() = 10,000
#0: ToddCoxeter: n = lookahead_next() = 5,000,000
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.2.66 | active | killed | defined
#0: ToddCoxeter: nodes | 10,644,480 | 52,857,227 | 63,501,707
#0: ToddCoxeter: diff 0.2.65 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 63,866,880 | 0 | 100.0%
#0: ToddCoxeter: diff 0.2.65 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.2 = 1min4s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min37s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 STOP (finished)
#0: ToddCoxeter: run 0 | lookahead | lookbehind | hlt | felsch
#0: ToddCoxeter: num. phases | 1 | 0 | 0 | 1
#0: ToddCoxeter: time spent in phases | 1min4s (29%) | - (0%) | - (0%) | 2min33s (70%)
#0: ToddCoxeter: phase 0.2 = 1min4s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min38s
The index of the subgroup is 10644480
The computed index is the same as the claimed index: \(10,644,480\).