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Mathieu group M11

The claims on this page come from 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 M11 defines a group of order \(7,920\).

The code

In libsemigroups_pybind11, the following script constructs the presentation for 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)^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

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.