\(2^{1+6}.L_3(2)\) as a subgroup of Held group He

The subgroup generators used on this page come from https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/.

Claim

\[ \mathrm{He} = \langle a, b \mid a^2 = b^7 = (ab)^{17} = [a, b]^6 = [a, b^3]^5 = [a, babab^{-1}abab] = (ab)^4ab^2ab^{-3}ababab^{-1}ab^3ab^{-2}ab^2 = 1 \rangle. \]

The generators are claimed to generate a maximal subgroup isomorphic to \(2^{1+6}.L_3(2)\). More precisely, if

\[ H = \langle ab^3,\ (pb^{-1}pb)^2(pbpb^{-1})^2,\ \text{ where }p=(ab^3)^4 \rangle, \]

then

\[ [\mathrm{He} : H] = 187,425. \]

On this page, we verify that the subgroup generated by these words has the claimed index.

The code

In libsemigroups_pybind11, the following script constructs the presentation for He, adds the generating pairs that define the maximal subgroup, and runs the Todd-Coxeter algorithm. The source code marks three of the presentation relations as redundant. We do not verify that these relations are redundant here.

Code

from libsemigroups_pybind11 import (
    Presentation,
    ToddCoxeter,
    congruence_kind,
    presentation,
)
from libsemigroups_pybind11.words import parse_relations as parse


# Setup the presentation object with the empty 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("a^2"), "")
presentation.add_rule(p, parse("b^7"), "")
presentation.add_rule(p, parse("(ab)^17"), "")
presentation.add_rule(p, parse("(ab^3)^8"), "")  # redundant
presentation.add_rule(p, parse("(ab^2ab^2aBBB)^3"), "")  # redundant
presentation.add_rule(p, parse("(a,babaBabab)"), "")
presentation.add_rule(p, parse("(a,b^3)^5"), "")
presentation.add_rule(p, parse("(a,b)^6"), "")
presentation.add_rule(
    p, parse("ab(abaBB)^2ab^2aBab^2aBB(abaB)^2"), ""
)  # redundant, maybe useful.
presentation.add_rule(p, parse("(ab)^4ab^2aB^3ababaBab^3aB^2ab^2"), "")

presentation.balance(p, "abAB", "ABab")
presentation.replace_subword(p, "A", "a")
p.alphabet("abB")

tc = ToddCoxeter(congruence_kind.onesided, p)
tc.strategy(ToddCoxeter.options.strategy.felsch)

# The subgroup generators here are taken from one of the straight line
# programmes from:
# https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/He/

# takes approx 4.5 minutes

p = parse("(ab^3)^4")
tc.add_generating_pair(parse("ab^3"), "")
tc.add_generating_pair(parse(f"({p}B{p}b)^2({p}b{p}B)^2"), "")
tc.use_relations_in_extra(True)

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| = 3, |R| = 14, |u| + |v| ∈ [2, 40], ∑(|u| + |v|) = 259
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 START
#0: ToddCoxeter: FELSCH 0.0.0     |        active |           killed |         defined
#0: ToddCoxeter: nodes            |           261 |               37 |             298
#0: ToddCoxeter:                  |        active |          missing |      % complete
#0: ToddCoxeter: edges            |           304 |              479 |           38.8%
#0: ToddCoxeter: time             | run 0 = 104µs | all runs = 104µs | elapsed = 342µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1       |         active |            killed |          defined
#1: ToddCoxeter: nodes              |      2,426,227 |            52,073 |        2,478,337
#1: ToddCoxeter: diff 0.0.0         |     +2,425,966 |           +52,036 |       +2,478,039
#1: ToddCoxeter:                    |         active |           missing |       % complete
#1: ToddCoxeter: edges              |      5,141,618 |         2,137,063 |            70.6%
#1: ToddCoxeter: diff 0.0.0         |     +5,141,314 |        +2,136,584 |           +31.8%
#1: ToddCoxeter: phase 0.0 = 1.003s | run 0 = 1.003s | all runs = 1.003s | elapsed = 1.003s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.2       |         active |            killed |          defined
#1: ToddCoxeter: nodes              |      4,813,554 |           108,463 |        4,922,040
#1: ToddCoxeter: diff 0.0.1         |     +2,387,327 |           +56,390 |       +2,443,703
#1: ToddCoxeter: diff 0.0.0         |     +4,813,293 |          +108,426 |       +4,921,742
#1: ToddCoxeter:                    |         active |           missing |       % complete
#1: ToddCoxeter: edges              |     10,203,878 |         4,236,784 |            70.7%
#1: ToddCoxeter: diff 0.0.1         |     +5,062,260 |        +2,099,721 |            +0.0%
#1: ToddCoxeter: diff 0.0.0         |    +10,203,574 |        +4,236,305 |           +31.8%
#1: ToddCoxeter: phase 0.0 = 2.004s | run 0 = 2.004s | all runs = 2.004s | elapsed = 2.004s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.3       |         active |            killed |          defined
#1: ToddCoxeter: nodes              |      7,279,489 |           169,041 |        7,448,552
#1: ToddCoxeter: diff 0.0.2         |     +2,465,935 |           +60,578 |       +2,526,512
#1: ToddCoxeter: diff 0.0.0         |     +7,279,228 |          +169,004 |       +7,448,254
#1: ToddCoxeter:                    |         active |           missing |       % complete
#1: ToddCoxeter: edges              |     15,432,998 |         6,405,469 |            70.7%
#1: ToddCoxeter: diff 0.0.2         |     +5,229,120 |        +2,168,685 |            +0.0%
#1: ToddCoxeter: diff 0.0.0         |    +15,432,694 |        +6,404,990 |           +31.8%
#1: ToddCoxeter: phase 0.0 = 3.009s | run 0 = 3.009s | all runs = 3.009s | elapsed = 3.009s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.4       |         active |            killed |          defined
#1: ToddCoxeter: nodes              |      9,597,950 |           232,139 |        9,830,108
#1: ToddCoxeter: diff 0.0.3         |     +2,318,461 |           +63,098 |       +2,381,556
#1: ToddCoxeter: diff 0.0.0         |     +9,597,689 |          +232,102 |       +9,829,810
#1: ToddCoxeter:                    |         active |           missing |       % complete
#1: ToddCoxeter: edges              |     20,349,896 |         8,443,954 |            70.7%
#1: ToddCoxeter: diff 0.0.3         |     +4,916,898 |        +2,038,485 |            +0.0%
#1: ToddCoxeter: diff 0.0.0         |    +20,349,592 |        +8,443,475 |           +31.8%
#1: ToddCoxeter: phase 0.0 = 4.014s | run 0 = 4.014s | all runs = 4.014s | elapsed = 4.014s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.5       |         active |            killed |          defined
#1: ToddCoxeter: nodes              |     12,053,029 |           297,869 |       12,350,930
#1: ToddCoxeter: diff 0.0.4         |     +2,455,079 |           +65,730 |       +2,520,822
#1: ToddCoxeter: diff 0.0.0         |    +12,052,768 |          +297,832 |      +12,350,632
#1: ToddCoxeter:                    |         active |           missing |       % complete
#1: ToddCoxeter: edges              |     25,556,108 |        10,602,979 |            70.7%
[... lines omitted ...]
#0: ToddCoxeter: LOOKAHEAD 0.1.1  |          active |             killed |           defined
#0: ToddCoxeter: nodes            |         187,425 |        513,212,960 |       513,400,385
#0: ToddCoxeter: diff 0.1.0       |              +0 |                 +0 |                +0
#0: ToddCoxeter:                  |          active |            missing |        % complete
#0: ToddCoxeter: edges            |         562,275 |                  0 |            100.0%
#0: ToddCoxeter: diff 0.1.0       |              +0 |                 +0 |             +0.0%
#0: ToddCoxeter: phase 0.1 = 74ms | run 0 = 4min37s | all runs = 4min37s | elapsed = 4min37s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 374,850, m = 10,000) (-4,625,150)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where:  a = number_of_nodes_active()     = 187,425
#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            |         187,425 |        513,212,960 |       513,400,385
#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            |         562,275 |                  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 = 74ms | run 0 = 4min37s | all runs = 4min37s | elapsed = 4min37s
++++++++++++++++++++++++++++++++
#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 |       74ms (0%) |             - (0%) |            - (0%) | 4min37s (100%)
#0: ToddCoxeter: phase 0.2 = 75ms     | run 0 = 4min37s | all runs = 4min37s | elapsed = 4min37s
The index of the subgroup is 187425

The computed index is the same as the claimed index: \(187,425\).