\(5^2:4A_4\) 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 \(5^2:4A_4\). More precisely, if
\[
H = \langle a,\ abab^2ab^{-3}ab \rangle,
\]
then
\[
[\mathrm{He} : H] = 3,358,656.
\]
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 54s
# These generators are from the webpages of the ATLAS
tc.add_generating_pair("a", "")
tc.add_generating_pair(parse("abab^2aB^3ab"), "")
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 | 10 | 0 | 10
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 12 | 18 | 40.0%
#0: ToddCoxeter: time | run 0 = 54µs | all runs = 54µs | elapsed = 231µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 2,432,372 | 62,383 | 2,494,757
#1: ToddCoxeter: diff 0.0.0 | +2,432,362 | +62,383 | +2,494,747
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 5,157,707 | 2,139,409 | 70.7%
#1: ToddCoxeter: diff 0.0.0 | +5,157,695 | +2,139,391 | +30.7%
#1: ToddCoxeter: phase 0.0 = 1.004s | run 0 = 1.004s | all runs = 1.004s | elapsed = 1.004s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined
#1: ToddCoxeter: nodes | 4,842,108 | 140,945 | 4,983,074
#1: ToddCoxeter: diff 0.0.1 | +2,409,736 | +78,562 | +2,488,317
#1: ToddCoxeter: diff 0.0.0 | +4,842,098 | +140,945 | +4,983,064
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 10,271,116 | 4,255,208 | 70.7%
#1: ToddCoxeter: diff 0.0.1 | +5,113,409 | +2,115,799 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +10,271,104 | +4,255,190 | +30.7%
#1: ToddCoxeter: phase 0.0 = 2.009s | run 0 = 2.009s | all runs = 2.009s | elapsed = 2.009s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined
#1: ToddCoxeter: nodes | 7,158,599 | 221,358 | 7,379,979
#1: ToddCoxeter: diff 0.0.2 | +2,316,491 | +80,413 | +2,396,905
#1: ToddCoxeter: diff 0.0.0 | +7,158,589 | +221,358 | +7,379,969
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 15,187,405 | 6,288,392 | 70.7%
#1: ToddCoxeter: diff 0.0.2 | +4,916,289 | +2,033,184 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +15,187,393 | +6,288,374 | +30.7%
#1: ToddCoxeter: phase 0.0 = 3.011s | run 0 = 3.011s | all runs = 3.011s | elapsed = 3.011s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.4 | active | killed | defined
#1: ToddCoxeter: nodes | 9,255,773 | 422,754 | 9,678,545
#1: ToddCoxeter: diff 0.0.3 | +2,097,174 | +201,396 | +2,298,566
#1: ToddCoxeter: diff 0.0.0 | +9,255,763 | +422,754 | +9,678,535
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 19,641,556 | 8,125,763 | 70.7%
#1: ToddCoxeter: diff 0.0.3 | +4,454,151 | +1,837,371 | +0.0%
#1: ToddCoxeter: diff 0.0.0 | +19,641,544 | +8,125,745 | +30.7%
#1: ToddCoxeter: phase 0.0 = 4.016s | run 0 = 4.016s | all runs = 4.016s | elapsed = 4.016s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.5 | active | killed | defined
#1: ToddCoxeter: nodes | 6,914,229 | 3,237,712 | 10,151,941
#1: ToddCoxeter: diff 0.0.4 | -2,341,544 | +2,814,958 | +473,396
#1: ToddCoxeter: diff 0.0.0 | +6,914,219 | +3,237,712 | +10,151,931
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 14,845,037 | 5,897,650 | 71.6%
[... lines omitted ...]
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 10,075,968 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.10 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 10.090s | run 0 = 54.796s | all runs = 54.796s | elapsed = 54.796s
#0: ToddCoxeter: lookahead_next() is now n x f = 10,000,000 (+5,000,000)
#0: ToddCoxeter: because: l < (l + a) / t = 839,664
#0: ToddCoxeter: where: a = number_of_nodes_active() = 3,358,656
#0: ToddCoxeter: f = lookahead_growth_factor() = 2
#0: ToddCoxeter: l = nodes killed in lookahead = 0
#0: ToddCoxeter: n = lookahead_next() = 5,000,000
#0: ToddCoxeter: t = lookahead_growth_threshold() = 4
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.2.12 | active | killed | defined
#0: ToddCoxeter: nodes | 3,358,656 | 7,414,887 | 10,773,543
#0: ToddCoxeter: diff 0.2.11 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 10,075,968 | 0 | 100.0%
#0: ToddCoxeter: diff 0.2.11 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.2 = 10.090s | run 0 = 54.796s | all runs = 54.796s | elapsed = 54.796s
++++++++++++++++++++++++++++++++
#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 | 10.090s (18%) | - (0%) | - (0%) | 44.705s (82%)
#0: ToddCoxeter: phase 0.2 = 10.147s | run 0 = 54.852s | all runs = 54.852s | elapsed = 54.852s
The index of the subgroup is 3358656
The computed index is the same as the claimed index: \(3,358,656\).