Tits group T
The claims on this page come from https://brauer.maths.qmul.ac.uk/Atlas/v3/exc/TF42/.
Claim
\[
\mathrm{T} = \langle a, b \mid a^2 = b^3 = (ab)^{13} = [a, b]^5 = [a, bab]^4 = ((ab)^4ab^{−1})^6 = 1 \rangle
\]
with
\[
|\mathrm{T}| = 17,971,200.
\]
On this page, we verify that the above claimed presentation of the Tits group T3 defines a group of order \(17,971,200\).
The code
In libsemigroups_pybind11, the following script constructs the presentation for T and runs the Todd-Coxeter algorithm.
Code
from libsemigroups_pybind11 import (
Presentation,
ToddCoxeter,
congruence_kind,
presentation,
)
from libsemigroups_pybind11.words import parse_relations
# Setup the presentation object with the empty word 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_relations("a^2"), "")
presentation.add_rule(p, parse_relations("b^3"), "")
presentation.add_rule(p, parse_relations("(ab)^13"), "")
presentation.add_rule(p, parse_relations("(a,b)^5"), "")
presentation.add_rule(p, parse_relations("(a,bab)^4"), "")
presentation.add_rule(p, parse_relations("(ababababaB)^6"), "")
presentation.balance(p, "abAB", "ABab")
presentation.replace_subword(p, "A", "a")
presentation.replace_subword(p, "B", "bb")
p.alphabet("ab")
tc = ToddCoxeter(congruence_kind.twosided, p)
tc.strategy(ToddCoxeter.options.strategy.felsch)
print(f"The size of the group 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| = 2, |R| = 10, |u| + |v| ∈ [2, 75], ∑(|u| + |v|) = 190
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 START
#0: ToddCoxeter: FELSCH 0.0.0 | active | killed | defined
#0: ToddCoxeter: nodes | 1 | 0 | 1
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 0 | 2 | 0.0%
#0: ToddCoxeter: time | run 0 = 28µs | all runs = 28µs | elapsed = 99µs
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#1: ToddCoxeter: nodes | 931,110 | 56,827 | 987,966
#1: ToddCoxeter: diff 0.0.0 | +931,109 | +56,827 | +987,965
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 1,636,776 | 225,444 | 87.9%
#1: ToddCoxeter: diff 0.0.0 | +1,636,776 | +225,442 | +87.9%
#1: ToddCoxeter: phase 0.0 = 1.000s | run 0 = 1.000s | all runs = 1.000s | elapsed = 1.000s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.2 | active | killed | defined
#1: ToddCoxeter: nodes | 1,689,659 | 107,031 | 1,796,700
#1: ToddCoxeter: diff 0.0.1 | +758,549 | +50,204 | +808,734
#1: ToddCoxeter: diff 0.0.0 | +1,689,658 | +107,031 | +1,796,699
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 2,979,269 | 400,049 | 88.2%
#1: ToddCoxeter: diff 0.0.1 | +1,342,493 | +174,605 | +0.3%
#1: ToddCoxeter: diff 0.0.0 | +2,979,269 | +400,047 | +88.2%
#1: ToddCoxeter: phase 0.0 = 2.000s | run 0 = 2.000s | all runs = 2.000s | elapsed = 2.000s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.3 | active | killed | defined
#1: ToddCoxeter: nodes | 2,422,503 | 156,074 | 2,578,589
#1: ToddCoxeter: diff 0.0.2 | +732,844 | +49,043 | +781,889
#1: ToddCoxeter: diff 0.0.0 | +2,422,502 | +156,074 | +2,578,588
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 4,279,477 | 565,529 | 88.3%
#1: ToddCoxeter: diff 0.0.2 | +1,300,208 | +165,480 | +0.2%
#1: ToddCoxeter: diff 0.0.0 | +4,279,477 | +565,527 | +88.3%
#1: ToddCoxeter: phase 0.0 = 3.001s | run 0 = 3.001s | all runs = 3.001s | elapsed = 3.001s
++++++++++++++++++++++++++++++++
#1: ToddCoxeter: FELSCH 0.0.4 | active | killed | defined
#1: ToddCoxeter: nodes | 3,109,354 | 202,556 | 3,311,914
#1: ToddCoxeter: diff 0.0.3 | +686,851 | +46,482 | +733,325
#1: ToddCoxeter: diff 0.0.0 | +3,109,353 | +202,556 | +3,311,913
#1: ToddCoxeter: | active | missing | % complete
#1: ToddCoxeter: edges | 5,499,715 | 718,993 | 88.4%
#1: ToddCoxeter: diff 0.0.3 | +1,220,238 | +153,464 | +0.1%
#1: ToddCoxeter: diff 0.0.0 | +5,499,715 | +718,991 | +88.4%
#1: ToddCoxeter: phase 0.0 = 4.001s | run 0 = 4.001s | all runs = 4.001s | elapsed = 4.001s
++++++++++++++++++++++++++++++++
[... lines omitted ...]
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.2.53 | active | killed | defined
#0: ToddCoxeter: nodes | 17,971,200 | 3,439,895 | 21,411,095
#0: ToddCoxeter: diff 0.2.52 | +0 | +0 | +0
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 35,942,400 | 0 | 100.0%
#0: ToddCoxeter: diff 0.2.52 | +0 | +0 | +0.0%
#0: ToddCoxeter: diff 0.2.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.2 = 51.384s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min37s
++++++++++++++++++++++++++++++++
#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 | 51.384s (24%) | - (0%) | - (0%) | 2min46s (76%)
#0: ToddCoxeter: phase 0.2 = 51.594s | run 0 = 3min37s | all runs = 3min37s | elapsed = 3min37s
The size of the group is 17971200
The computed size of the group matches the size of the group provided on the ATLAS: \(17,971,200\).