\(2^6:3.S_6\) (I) 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 \(2^6:3.S_6\). More precisely, if
\[
H = \langle a,\ ab^{-2}ab^3ab^{-3}ab^{-1} \rangle,
\]
then
\[
[\mathrm{He} : H] = 29,155.
\]
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 20s
tc.add_generating_pair("a", "")
tc.add_generating_pair("aBBabbbaBBBaB", "")
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 | 12 | 0 | 12
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 14 | 22 | 38.9%
#0: ToddCoxeter: time | run 0 = 59µs | all runs = 59µs | elapsed = 110µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 2,473,292 | 58,553 | 2,531,861
#1: ToddCoxeter: diff 0.0.0 | +2,473,280 | +58,553 | +2,531,849
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 5,243,526 | 2,176,350 | 70.7%
#1: ToddCoxeter: diff 0.0.0 | +5,243,512 | +2,176,328 | +31.8%
#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 | 4,907,283 | 123,428 | 5,030,726
#1: ToddCoxeter: diff 0.0.1 | +2,433,991 | +64,875 | +2,498,865
#1: ToddCoxeter: diff 0.0.0 | +4,907,271 | +123,428 | +5,030,714
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 10,405,718 | 4,316,131 | 70.7%
#1: ToddCoxeter: diff 0.0.1 | +5,162,192 | +2,139,781 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +10,405,704 | +4,316,109 | +31.8%
#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 | 7,356,730 | 202,010 | 7,558,764
#1: ToddCoxeter: diff 0.0.2 | +2,449,447 | +78,582 | +2,528,038
#1: ToddCoxeter: diff 0.0.0 | +7,356,718 | +202,010 | +7,558,752
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 15,602,237 | 6,467,953 | 70.7%
#1: ToddCoxeter: diff 0.0.2 | +5,196,519 | +2,151,822 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +15,602,223 | +6,467,931 | +31.8%
#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 | 9,645,930 | 298,259 | 9,944,212
#1: ToddCoxeter: diff 0.0.3 | +2,289,200 | +96,249 | +2,385,448
#1: ToddCoxeter: diff 0.0.0 | +9,645,918 | +298,259 | +9,944,200
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 20,461,214 | 8,476,576 | 70.7%
#1: ToddCoxeter: diff 0.0.3 | +4,858,977 | +2,008,623 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +20,461,200 | +8,476,554 | +31.8%
#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 | 12,015,424 | 393,537 | 12,408,973
#1: ToddCoxeter: diff 0.0.4 | +2,369,494 | +95,278 | +2,464,761
#1: ToddCoxeter: diff 0.0.0 | +12,015,412 | +393,537 | +12,408,961
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 25,491,801 | 10,554,471 | 70.7%
[... lines omitted ...]
#0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined
#0: ToddCoxeter: nodes | 29,155 | 44,559,134 | 44,588,289
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 87,465 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 8ms | run 0 = 20.191s | all runs = 20.191s | elapsed = 20.192s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 58,310, m = 10,000) (-4,941,690)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 29,155
#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 | 29,155 | 44,559,134 | 44,588,289
#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 | 87,465 | 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 = 8ms | run 0 = 20.191s | all runs = 20.191s | elapsed = 20.192s
++++++++++++++++++++++++++++++++
#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 | 8ms (0%) | - (0%) | - (0%) | 20.183s (100%)
#0: ToddCoxeter: phase 0.2 = 8ms | run 0 = 20.192s | all runs = 20.192s | elapsed = 20.192s
The index of the subgroup is 29155
The computed index is the same as the claimed index: \(29,155\).