Mathieu group M11 as a subgroup of Mathieu group M23 (Presentation 2)
The claims on this page come from https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1.
Claim
\[
\mathrm{M}_{23} = \langle a, b \mid a^2 = b^4 = (ab^2)^6 = (abab^{−1}ab^2)^4 = abababab^{−1}ab^2abab^{−1}abab^{−1}abababab^{−1}ab^{−1}ab^{−1} = abab^2abab^2abab^2ab^2ab^{−1}ab^2ab^{−1}abab^2abab^{−1}ab^2 = abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2ab^{−1}ab^2ab^2ab^{−1}ab^2ab^2 = ababababab^2abab^{−1}abab^2ababab^{−1}ab^2abab^2ab^2abab^{−1}ab^{−1}abab^2 = 1 \rangle
\]
and
\[
\mathrm{M}_{11} = {\mathrm{M}_{23}} \mathbin{/} \langle a, b \mid b = bbabbababbbabbabab = 1 \rangle
\]
with
\[
[\mathrm{M}_{23} : \mathrm{M}_{11}] = 23.
\]
The code
In libsemigroups_pybind11, the following script constructs the presentation for M23, adds the generating pairs that define the subgroup M11, 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^4"), "")
presentation.add_rule(p, parse_relations("(ab^2)^6"), "")
presentation.add_rule(p, parse_relations("(a,b)^6"), "")
presentation.add_rule(p, parse_relations("(abaBab^2)^4"), "")
presentation.add_rule(p, parse_relations("abababaBab^2abaBabaBabababaBaBaB"), "")
presentation.add_rule(p, parse_relations("abab^2abab^2abab^2ab^2aBab^2aBabab^2abaBab^2"), "")
presentation.add_rule(
p, parse_relations("abab^2ab^2abab^2ab^2abab^2ab^2abab^2ab^2aBab^2ab^2aBab^2ab^2"), ""
)
presentation.add_rule(
p, parse_relations("ababababab^2abaBabab^2ababaBab^2abab^2ab^2abaBaBabab^2"), ""
)
# Add generating pairs for the maximal subgroup
tc = ToddCoxeter(congruence_kind.onesided, p)
tc.add_generating_pair("b", "")
tc.add_generating_pair("bbabbababbbabbabab", "")
# 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
Output from the Python script
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
#0: ToddCoxeter: |A| = 4, |R| = 13, |u| + |v| ∈ [2, 48], ∑(|u| + |v|) = 248
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 START
#0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined
#0: ToddCoxeter: nodes | 149 | 68 | 217
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 221 | 375 | 37.1%
#0: ToddCoxeter: time | run 0 = 390µs | all runs = 390µs | elapsed = 457µs
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 STOP
#0: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#0: ToddCoxeter: nodes | 1,288 | 25,400 | 26,688
#0: ToddCoxeter: diff 0.0.0 | +1,139 | +25,332 | +26,471
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 5,152 | 0 | 100.0%
#0: ToddCoxeter: diff 0.0.0 | +4,931 | -375 | +62.9%
#0: ToddCoxeter: phase 0.0 = 33ms | run 0 = 33ms | all runs = 33ms | elapsed = 33ms
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined
#0: ToddCoxeter: nodes | 1,288 | 25,400 | 26,688
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 5,152 | 0 | 100.0%
#0: ToddCoxeter: time | run 0 = 33ms | all runs = 33ms | elapsed = 33ms
#0: ToddCoxeter: triggered because there are skipped definitions (1,288 active nodes)!
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 STOP
#0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined
#0: ToddCoxeter: nodes | 1,288 | 25,400 | 26,688
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 5,152 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 474µs | run 0 = 33ms | all runs = 33ms | elapsed = 33ms
#0: ToddCoxeter: lookahead_next() is now max(f x a = 2,576, m = 10,000) (-4,990,000)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 1,288
#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 0.2.2 | active | killed | defined
#0: ToddCoxeter: nodes | 1,288 | 25,400 | 26,688
#0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 5,152 | 0 | 100.0%
#0: ToddCoxeter: diff 0.2.1 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.2 = 498µs | run 0 = 33ms | all runs = 33ms | elapsed = 33ms
++++++++++++++++++++++++++++++++
#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 | 495µs (1%) | - (0%) | - (0%) | 33ms (97%)
#0: ToddCoxeter: phase 0.2 = 519µs | run 0 = 33ms | all runs = 33ms | elapsed = 34ms
The index of the subgroup is 1288
The computed index is the same as the claimed index: \(1,288\).