\(S_4 \times L_3(2)\) 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 \(S_4 \times L_3(2)\). More precisely, if
\[
H = \langle (q^{-1})^2(ab^2)^6q^2,\ (ab)^6(ab^2)^4(b^{-1}a)^6,\ \text{ where }q=ababb \rangle,
\]
then
\[
[\mathrm{He} : H] = 999,600.
\]
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 2.5 minutes
q = "ababb"
Q = ("".join(reversed(q))).replace("b", "B")
tc.add_generating_pair(parse(f"({Q})^2(ab^2)^6({q})^2"), "")
tc.add_generating_pair(parse("(ab)^6(ab^2)^4(Ba)^6"), "")
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 | 48 | 25 | 73
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 74 | 70 | 51.4%
#0: ToddCoxeter: time | run 0 = 70µs | all runs = 70µs | elapsed = 264µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 2,473,159 | 53,957 | 2,527,133
#1: ToddCoxeter: diff 0.0.0 | +2,473,111 | +53,932 | +2,527,060
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 5,242,117 | 2,177,360 | 70.7%
#1: ToddCoxeter: diff 0.0.0 | +5,242,043 | +2,177,290 | +19.3%
#1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined
#1: ToddCoxeter: nodes | 4,992,438 | 112,926 | 5,105,379
#1: ToddCoxeter: diff 0.0.1 | +2,519,279 | +58,969 | +2,578,246
#1: ToddCoxeter: diff 0.0.0 | +4,992,390 | +112,901 | +5,105,306
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 10,583,240 | 4,394,074 | 70.7%
#1: ToddCoxeter: diff 0.0.1 | +5,341,123 | +2,216,714 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +10,583,166 | +4,394,004 | +19.3%
#1: ToddCoxeter: phase 0.0 = 2.010s | run 0 = 2.010s | all runs = 2.010s | elapsed = 2.010s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined
#1: ToddCoxeter: nodes | 7,553,726 | 174,059 | 7,727,802
#1: ToddCoxeter: diff 0.0.2 | +2,561,288 | +61,133 | +2,622,423
#1: ToddCoxeter: diff 0.0.0 | +7,553,678 | +174,034 | +7,727,729
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 16,014,216 | 6,646,962 | 70.7%
#1: ToddCoxeter: diff 0.0.2 | +5,430,976 | +2,252,888 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +16,014,142 | +6,646,892 | +19.3%
#1: ToddCoxeter: phase 0.0 = 3.015s | run 0 = 3.015s | all runs = 3.015s | elapsed = 3.015s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.4 | active | killed | defined
#1: ToddCoxeter: nodes | 10,022,736 | 236,866 | 10,259,624
#1: ToddCoxeter: diff 0.0.3 | +2,469,010 | +62,807 | +2,531,822
#1: ToddCoxeter: diff 0.0.0 | +10,022,688 | +236,841 | +10,259,551
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 21,250,224 | 8,817,984 | 70.7%
#1: ToddCoxeter: diff 0.0.3 | +5,236,008 | +2,171,022 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +21,250,150 | +8,817,914 | +19.3%
#1: ToddCoxeter: phase 0.0 = 4.020s | run 0 = 4.020s | all runs = 4.020s | elapsed = 4.020s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.5 | active | killed | defined
#1: ToddCoxeter: nodes | 12,591,904 | 299,790 | 12,891,716
#1: ToddCoxeter: diff 0.0.4 | +2,569,168 | +62,924 | +2,632,092
#1: ToddCoxeter: diff 0.0.0 | +12,591,856 | +299,765 | +12,891,643
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 26,698,162 | 11,077,550 | 70.7%
[... lines omitted ...]
#0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 2,998,800 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.1 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 1.339s | run 0 = 2min31s | all runs = 2min31s | elapsed = 2min31s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 1,999,200, m = 10,000) (-3,000,800)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 999,600
#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 0.2.3 | active | killed | defined
#0: ToddCoxeter: nodes | 999,600 | 287,397,473 | 288,397,073
#0: ToddCoxeter: diff 0.2.2 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 2,998,800 | 0 | 100.0%
#0: ToddCoxeter: diff 0.2.2 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.2 = 1.339s | run 0 = 2min31s | all runs = 2min31s | elapsed = 2min31s
++++++++++++++++++++++++++++++++
#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 | 1.339s (1%) | - (0%) | - (0%) | 2min29s (99%)
#0: ToddCoxeter: phase 0.2 = 1.346s | run 0 = 2min31s | all runs = 2min31s | elapsed = 2min31s
The index of the subgroup is 999600
The computed index is the same as the claimed index: \(999,600\).