\(7^{1+2}:(3 \times S_3)\) 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 \(7^{1+2}:(3 \times S_3)\). More precisely, if
\[
H = \langle (b^{-1}a)^7(ab^2)^6(ab)^7,\ (q^{-1})^5(ab^2)^2q^5,\ \text{ where }q=ababb \rangle,
\]
then
\[
[\mathrm{He} : H] = 652,800.
\]
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("(Ba)^7(ab^2)^6(ab)^7"), "")
tc.add_generating_pair(parse(f"({Q})^5(ab^2)^2({q})^5"), "")
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 | 57 | 43 | 100
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 100 | 71 | 58.5%
#0: ToddCoxeter: time | run 0 = 81µs | all runs = 81µs | elapsed = 302µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 2,498,917 | 52,325 | 2,551,269
#1: ToddCoxeter: diff 0.0.0 | +2,498,860 | +52,282 | +2,551,169
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 5,296,218 | 2,200,533 | 70.6%
#1: ToddCoxeter: diff 0.0.0 | +5,296,118 | +2,200,462 | +12.2%
#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 | 4,989,394 | 110,077 | 5,099,485
#1: ToddCoxeter: diff 0.0.1 | +2,490,477 | +57,752 | +2,548,216
#1: ToddCoxeter: diff 0.0.0 | +4,989,337 | +110,034 | +5,099,385
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 10,576,096 | 4,392,086 | 70.7%
#1: ToddCoxeter: diff 0.0.1 | +5,279,878 | +2,191,553 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +10,575,996 | +4,392,015 | +12.2%
#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 | 7,541,248 | 170,690 | 7,711,948
#1: ToddCoxeter: diff 0.0.2 | +2,551,854 | +60,613 | +2,612,463
#1: ToddCoxeter: diff 0.0.0 | +7,541,191 | +170,647 | +7,711,848
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 15,986,986 | 6,636,758 | 70.7%
#1: ToddCoxeter: diff 0.0.2 | +5,410,890 | +2,244,672 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +15,986,886 | +6,636,687 | +12.2%
#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 | 9,990,510 | 226,874 | 10,217,405
#1: ToddCoxeter: diff 0.0.3 | +2,449,262 | +56,184 | +2,505,457
#1: ToddCoxeter: diff 0.0.0 | +9,990,453 | +226,831 | +10,217,305
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 21,180,244 | 8,791,286 | 70.7%
#1: ToddCoxeter: diff 0.0.3 | +5,193,258 | +2,154,528 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +21,180,144 | +8,791,215 | +12.2%
#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 | 12,531,004 | 287,403 | 12,818,441
#1: ToddCoxeter: diff 0.0.4 | +2,540,494 | +60,529 | +2,601,036
#1: ToddCoxeter: diff 0.0.0 | +12,530,947 | +287,360 | +12,818,341
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 26,567,136 | 11,025,876 | 70.7%
[... lines omitted ...]
#0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined
#0: ToddCoxeter: nodes | 652,800 | 251,782,228 | 252,435,028
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 1,958,400 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 572ms | run 0 = 2min24s | all runs = 2min24s | elapsed = 2min24s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 1,305,600, m = 10,000) (-3,694,400)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 652,800
#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 | 652,800 | 251,782,228 | 252,435,028
#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 | 1,958,400 | 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 = 572ms | run 0 = 2min24s | all runs = 2min24s | elapsed = 2min24s
++++++++++++++++++++++++++++++++
#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 | 572ms (0%) | - (0%) | - (0%) | 2min23s (100%)
#0: ToddCoxeter: phase 0.2 = 576ms | run 0 = 2min24s | all runs = 2min24s | elapsed = 2min24s
The index of the subgroup is 652800
The computed index is the same as the claimed index: \(652,800\).