\(2^6:3.S_6\) (II) 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
\[
H = \langle a,\ ab^{-2}ab^2ab^{-3}ab^{-1} \rangle,
\]
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 5s
tc.add_generating_pair("a", "")
tc.add_generating_pair("aBBabbaBBBaB", "")
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 | 11 | 0 | 11
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 13 | 20 | 39.4%
#0: ToddCoxeter: time | run 0 = 55µs | all runs = 55µs | elapsed = 103µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 2,450,610 | 56,933 | 2,507,577
#1: ToddCoxeter: diff 0.0.0 | +2,450,599 | +56,933 | +2,507,566
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 5,195,501 | 2,156,329 | 70.7%
#1: ToddCoxeter: diff 0.0.0 | +5,195,488 | +2,156,309 | +31.3%
#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 | 4,857,043 | 122,296 | 4,979,354
#1: ToddCoxeter: diff 0.0.1 | +2,406,433 | +65,363 | +2,471,777
#1: ToddCoxeter: diff 0.0.0 | +4,857,032 | +122,296 | +4,979,343
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 10,299,694 | 4,271,435 | 70.7%
#1: ToddCoxeter: diff 0.0.1 | +5,104,193 | +2,115,106 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +10,299,681 | +4,271,415 | +31.3%
#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 | 7,241,525 | 226,832 | 7,468,390
#1: ToddCoxeter: diff 0.0.2 | +2,384,482 | +104,536 | +2,489,036
#1: ToddCoxeter: diff 0.0.0 | +7,241,514 | +226,832 | +7,468,379
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 15,360,734 | 6,363,841 | 70.7%
#1: ToddCoxeter: diff 0.0.2 | +5,061,040 | +2,092,406 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +15,360,721 | +6,363,821 | +31.3%
#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 | 9,527,319 | 309,104 | 9,836,440
#1: ToddCoxeter: diff 0.0.3 | +2,285,794 | +82,272 | +2,368,050
#1: ToddCoxeter: diff 0.0.0 | +9,527,308 | +309,104 | +9,836,429
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 20,212,108 | 8,369,849 | 70.7%
#1: ToddCoxeter: diff 0.0.3 | +4,851,374 | +2,006,008 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +20,212,095 | +8,369,829 | +31.3%
#1: ToddCoxeter: phase 0.0 = 4.020s | run 0 = 4.020s | all runs = 4.020s | elapsed = 4.020s
#0: ToddCoxeter: large collapse, number of coincidences 100,000 >= 100,000 = large_collapse()!
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.5 | active | killed | defined
#1: ToddCoxeter: nodes | 45,650 | 11,465,859 | 11,511,509
#1: ToddCoxeter: diff 0.0.4 | -9,481,669 | +11,156,755 | +1,675,069
#1: ToddCoxeter: diff 0.0.0 | +45,639 | +11,465,859 | +11,511,498
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 70,390 | 66,560 | 51.4%
#1: ToddCoxeter: diff 0.0.4 | -20,141,718 | -8,303,289 | -19.3%
#1: ToddCoxeter: diff 0.0.0 | +70,377 | +66,540 | +12.0%
#1: ToddCoxeter: phase 0.0 = 5.025s | run 0 = 5.025s | all runs = 5.025s | elapsed = 5.025s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 STOP
#0: ToddCoxeter: FELSCH 0.0.6 | active | killed | defined
#0: ToddCoxeter: nodes | 29,155 | 11,482,367 | 11,511,522
#0: ToddCoxeter: diff 0.0.5 | -16,495 | +16,508 | +13
#0: ToddCoxeter: diff 0.0.0 | +29,144 | +11,482,367 | +11,511,511
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 87,465 | 0 | 100.0%
#0: ToddCoxeter: diff 0.0.5 | +17,075 | -66,560 | +48.6%
#0: ToddCoxeter: diff 0.0.0 | +87,452 | -20 | +60.6%
#0: ToddCoxeter: phase 0.0 = 5.029s | run 0 = 5.029s | all runs = 5.029s | elapsed = 5.029s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined
#0: ToddCoxeter: nodes | 29,155 | 11,482,367 | 11,511,522
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 87,465 | 0 | 100.0%
#0: ToddCoxeter: time | run 0 = 5.029s | all runs = 5.029s | elapsed = 5.029s
#0: ToddCoxeter: triggered because there are skipped definitions (29,155 active nodes)!
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 STOP
#0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined
#0: ToddCoxeter: nodes | 29,155 | 11,482,367 | 11,511,522
#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 = 9ms | run 0 = 5.038s | all runs = 5.038s | elapsed = 5.039s
#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 | 11,482,367 | 11,511,522
#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 = 9ms | run 0 = 5.039s | all runs = 5.039s | elapsed = 5.039s
++++++++++++++++++++++++++++++++
#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 | 9ms (0%) | - (0%) | - (0%) | 5.029s (100%)
#0: ToddCoxeter: phase 0.2 = 9ms | run 0 = 5.039s | all runs = 5.039s | elapsed = 5.039s
The index of the subgroup is 29155
The computed index is the same as the claimed index: \(29,155\).