Mathieu group M₂₂ 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

\[ \mathrm{M}_{22} = {\mathrm{M}_{23}} \mathbin{/} \langle a, b \mid a = bababbab = 1 \rangle \]

with

\[ [\mathrm{M}_{23} : \mathrm{M}_{22}] = 23. \]

Warning

The word \(bababbab\) in plain text on the page

https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/M23G1-max1W1

does not seem to generate a subgroup of the correct index. Expanding the GAP straightline programme at:

http://atlas.math.rwth-aachen.de/Atlas/v3/scripts/slp2.php%3Fformat=GAP&filename=M23G1-max1W1&filename_path=spor%252FM23&sub_dir=words

yields the word \(bababbbab\) which does generate a subgroup of the correct index.

Plain textStraightline program

In libsemigroups_pybind11, the following script constructs the presentation for M₂₃, adds the generating pairs that define the subgroup M₂₂, and runs the Todd-Coxeter algorithm. This is based on the plain text implementation from the ATLAS.

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("a", "")
tc.add_generating_pair("bababbab", "")

# 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()}")

In libsemigroups_pybind11, the following script constructs the presentation for M₂₃, adds the generating pairs that define the subgroup M₂₂, and runs the Todd-Coxeter algorithm. This is based on the GAP straightline program hosted on the ATLAS.

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("a", "")
tc.add_generating_pair("bababbbab", "")

# 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

Plain textPlain text

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            |           153 |               53 |             206
#0: ToddCoxeter:                  |        active |          missing |      % complete
#0: ToddCoxeter: edges            |           231 |              381 |           37.7%
#0: ToddCoxeter: time             | run 0 = 349µs | all runs = 349µs | elapsed = 420µs
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 STOP
#0: ToddCoxeter: FELSCH 0.0.1     |       active |          killed |        defined
#0: ToddCoxeter: nodes            |        1,288 |          14,407 |         15,695
#0: ToddCoxeter: diff 0.0.0       |       +1,135 |         +14,354 |        +15,489
#0: ToddCoxeter:                  |       active |         missing |     % complete
#0: ToddCoxeter: edges            |        5,152 |               0 |         100.0%
#0: ToddCoxeter: diff 0.0.0       |       +4,921 |            -381 |         +62.3%
#0: ToddCoxeter: phase 0.0 = 51ms | run 0 = 51ms | all runs = 51ms | elapsed = 51ms
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.1.2     |       active |          killed |        defined
#0: ToddCoxeter: nodes            |        1,288 |          14,407 |         15,695
#0: ToddCoxeter: diff 0.1.1       |           +0 |              +0 |             +0
#0: ToddCoxeter: diff 0.1.0       |       +1,135 |         +14,354 |        +15,489
#0: ToddCoxeter:                  |       active |         missing |     % complete
#0: ToddCoxeter: edges            |        5,152 |               0 |         100.0%
#0: ToddCoxeter: diff 0.1.1       |           +0 |              +0 |          +0.0%
#0: ToddCoxeter: diff 0.1.0       |       +4,921 |            -381 |         +62.3%
#0: ToddCoxeter: phase 0.1 = 51ms | run 0 = 51ms | all runs = 51ms | elapsed = 51ms
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 STOP (finished)
#0: ToddCoxeter: run 0                |    lookahead |      lookbehind |            hlt |       felsch
#0: ToddCoxeter: num. phases          |            0 |               0 |              0 |            1
#0: ToddCoxeter: time spent in phases |       - (0%) |          - (0%) |         - (0%) |   51ms (99%)
#0: ToddCoxeter: phase 0.1 = 51ms     | run 0 = 51ms | all runs = 51ms | elapsed = 51ms
The index of the subgroup is 1288

The computed index is \(1,288\) which is not the same as the claimed index: \(23\).

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            |           144 |               63 |           207
#0: ToddCoxeter:                  |        active |          missing |    % complete
#0: ToddCoxeter: edges            |           212 |              364 |         36.8%
#0: ToddCoxeter: time             | run 0 = 394µs | all runs = 394µs | elapsed = 2ms
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 STOP
#0: ToddCoxeter: FELSCH 0.0.1     |       active |          killed |        defined
#0: ToddCoxeter: nodes            |           23 |          40,048 |         40,071
#0: ToddCoxeter: diff 0.0.0       |         -121 |         +39,985 |        +39,864
#0: ToddCoxeter:                  |       active |         missing |     % complete
#0: ToddCoxeter: edges            |           92 |               0 |         100.0%
#0: ToddCoxeter: diff 0.0.0       |         -120 |            -364 |         +63.2%
#0: ToddCoxeter: phase 0.0 = 52ms | run 0 = 53ms | all runs = 53ms | elapsed = 54ms
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.1.0  |       active |          killed |        defined
#0: ToddCoxeter: nodes            |           23 |          40,048 |         40,071
#0: ToddCoxeter:                  |       active |         missing |     % complete
#0: ToddCoxeter: edges            |           92 |               0 |         100.0%
#0: ToddCoxeter: time             | run 0 = 53ms | all runs = 53ms | elapsed = 54ms
#0: ToddCoxeter: triggered because there are skipped definitions (23 active nodes)!
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 STOP
#0: ToddCoxeter: LOOKAHEAD 0.1.1  |       active |          killed |        defined
#0: ToddCoxeter: nodes            |           23 |          40,048 |         40,071
#0: ToddCoxeter: diff 0.1.0       |           +0 |              +0 |             +0
#0: ToddCoxeter:                  |       active |         missing |     % complete
#0: ToddCoxeter: edges            |           92 |               0 |         100.0%
#0: ToddCoxeter: diff 0.1.0       |           +0 |              +0 |          +0.0%
#0: ToddCoxeter: phase 0.1 = 31µs | run 0 = 53ms | all runs = 53ms | elapsed = 54ms
#0: ToddCoxeter: lookahead_next() is now max(f x a = 46, m = 10,000) (-4,990,000)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where:  a = number_of_nodes_active()     = 23
#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            |           23 |          40,048 |         40,071
#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            |           92 |               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 = 59µs | run 0 = 53ms | all runs = 53ms | elapsed = 54ms
++++++++++++++++++++++++++++++++
#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 |    54µs (0%) |          - (0%) |         - (0%) |   52ms (99%)
#0: ToddCoxeter: phase 0.2 = 83µs     | run 0 = 53ms | all runs = 53ms | elapsed = 55ms
The index of the subgroup is 23

The computed index is the same as the claimed index: \(23\).

Mathieu group M22 as a subgroup of Mathieu group M23 (Presentation 2)

Claim

The output

Mathieu group M₂₂ as a subgroup of Mathieu group M₂₃ (Presentation 2)