\(S_4(4):2\) as a subgroup of Held group He
The subgroup generators used on this page come from https://brauer.maths.qmul.ac.uk/Atlas/spor/He/mag/HeG1-P1.M.
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 \(S_4(4):2\). More precisely, if
\[
H = \langle a,\ b^{-2}ab^2,\ b^2ab^{-2} \rangle,
\]
then
\[
[\mathrm{He} : H] = 2,058.
\]
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)
# takes approx 1.6s
# The generators for this subgroup are taken from:
# https://brauer.maths.qmul.ac.uk/Atlas/spor/He/mag/HeG1-P1.M
# The generators on:
# https://brauer.maths.qmul.ac.uk/Atlas/v3/subgroup/HeG1-max1W1
# are:
# a, (abb)-7(bababababbababb)7(abb)7
# tc.add_generating_pair(parse("(BBa)^7(bababababbababb)^7(abb)^7"), "")
# and in the straightline programme at:
# https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max1W1
# are:
# a, and (abababbab)^10
# which is altogether rather confusing...
tc.add_generating_pair("a", "")
tc.add_generating_pair("BBabb", "")
tc.add_generating_pair("bbaBB", "")
print(f"The index of the subgroup is {tc.number_of_classes()}")
The output
The output of the enumeration is below:
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 | 5 | 4 | 9
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 11 | 4 | 73.3%
#0: ToddCoxeter: time | run 0 = 63µs | all runs = 63µs | elapsed = 111µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 2,389,517 | 86,354 | 2,475,902
#1: ToddCoxeter: diff 0.0.0 | +2,389,512 | +86,350 | +2,475,893
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 5,071,468 | 2,097,083 | 70.7%
#1: ToddCoxeter: diff 0.0.0 | +5,071,457 | +2,097,079 | -2.6%
#1: ToddCoxeter: phase 0.0 = 1.002s | run 0 = 1.002s | all runs = 1.002s | elapsed = 1.002s
#0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()!
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 STOP
#0: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined
#0: ToddCoxeter: nodes | 2,058 | 3,621,394 | 3,623,452
#0: ToddCoxeter: diff 0.0.1 | -2,387,459 | +3,535,040 | +1,147,550
#0: ToddCoxeter: diff 0.0.0 | +2,053 | +3,621,390 | +3,623,443
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 6,174 | 0 | 100.0%
#0: ToddCoxeter: diff 0.0.1 | -5,065,294 | -2,097,083 | +29.3%
#0: ToddCoxeter: diff 0.0.0 | +6,163 | -4 | +26.7%
#0: ToddCoxeter: phase 0.0 = 1.564s | run 0 = 1.564s | all runs = 1.564s | elapsed = 1.564s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined
#0: ToddCoxeter: nodes | 2,058 | 3,621,394 | 3,623,452
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 6,174 | 0 | 100.0%
#0: ToddCoxeter: time | run 0 = 1.564s | all runs = 1.564s | elapsed = 1.564s
#0: ToddCoxeter: triggered because there are skipped definitions (2,058 active nodes)!
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 STOP
#0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined
#0: ToddCoxeter: nodes | 2,058 | 3,621,394 | 3,623,452
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 6,174 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 412µs | run 0 = 1.565s | all runs = 1.565s | elapsed = 1.565s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 4,116, m = 10,000) (-4,990,000)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 2,058
#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 | 2,058 | 3,621,394 | 3,623,452
#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 | 6,174 | 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 = 435µs | run 0 = 1.565s | all runs = 1.565s | elapsed = 1.565s
++++++++++++++++++++++++++++++++
#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 | 433µs (0%) | - (0%) | - (0%) | 1.564s (100%)
#0: ToddCoxeter: phase 0.2 = 456µs | run 0 = 1.565s | all runs = 1.565s | elapsed = 1.565s
The index of the subgroup is 2058
The computed index is the same as the claimed index: \(2,058\).