\(7^2:2.L_2(7)\) 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/words/HeG1-max6W1.
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 \(7^2:2.L_2(7)\). More precisely, if
\[
H = \langle (ab)^5b^2a,\ b \rangle,
\]
then
\[
[\mathrm{He} : H] = 244,800.
\]
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)
# takes approx 1.6s
# The words are from:
# https://brauer.maths.qmul.ac.uk/Atlas/spor/He/words/HeG1-max6W1
tc.add_generating_pair(parse("(ab)^5b^2a"), "")
tc.add_generating_pair("b", "")
print(f"The by 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 | 12 | 1 | 13
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 16 | 20 | 44.4%
#0: ToddCoxeter: time | run 0 = 104µs | all runs = 104µs | elapsed = 314µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 999,860 | 1,081,844 | 2,081,604
#1: ToddCoxeter: diff 0.0.0 | +999,848 | +1,081,843 | +2,081,591
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 2,169,279 | 830,301 | 72.3%
#1: ToddCoxeter: diff 0.0.0 | +2,169,263 | +830,281 | +27.9%
#1: ToddCoxeter: phase 0.0 = 1.005s | run 0 = 1.005s | all runs = 1.005s | elapsed = 1.005s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 STOP
#0: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined
#0: ToddCoxeter: nodes | 244,800 | 1,839,761 | 2,084,561
#0: ToddCoxeter: diff 0.0.1 | -755,060 | +757,917 | +2,957
#0: ToddCoxeter: diff 0.0.0 | +244,788 | +1,839,760 | +2,084,548
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 734,400 | 0 | 100.0%
#0: ToddCoxeter: diff 0.0.1 | -1,434,879 | -830,301 | +27.7%
#0: ToddCoxeter: diff 0.0.0 | +734,384 | -20 | +55.6%
#0: ToddCoxeter: phase 0.0 = 1.515s | run 0 = 1.516s | all runs = 1.516s | elapsed = 1.516s
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined
#0: ToddCoxeter: nodes | 244,800 | 1,839,761 | 2,084,561
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 734,400 | 0 | 100.0%
#0: ToddCoxeter: time | run 0 = 1.516s | all runs = 1.516s | elapsed = 1.516s
#0: ToddCoxeter: triggered because there are skipped definitions (244,800 active nodes)!
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 STOP
#0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined
#0: ToddCoxeter: nodes | 244,800 | 1,839,761 | 2,084,561
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 734,400 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 100ms | run 0 = 1.616s | all runs = 1.616s | elapsed = 1.616s
#0: ToddCoxeter: lookahead_next() is now max(f x a = 489,600, m = 10,000) (-4,510,400)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 244,800
#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 | 244,800 | 1,839,761 | 2,084,561
#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 | 734,400 | 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 = 100ms | run 0 = 1.616s | all runs = 1.616s | elapsed = 1.616s
++++++++++++++++++++++++++++++++
#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 | 100ms (6%) | - (0%) | - (0%) | 1.515s (94%)
#0: ToddCoxeter: phase 0.2 = 101ms | run 0 = 1.617s | all runs = 1.617s | elapsed = 1.617s
The by index of the subgroup is 244800
The computed index is the same as the claimed index: \(244,800\).