Janko group J1
The claims on this page come from https://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J1/.
Claim
\[
\mathrm{J}_1 = \langle a, b \mid a^2 = b^3 = (ab)^7 = (ab(abab^{−1})^3)^5 = (ab(abab^{−1})^6abab(ab^{−1})^2)^2 = 1 \rangle
\]
with
\[
|\mathrm{J}_1| = 175,560.
\]
On this page, we verify that the above claimed presentation of the Janko group J1 defines a group of order \(175,560\).
The code
In libsemigroups_pybind11, the following script constructs the presentation for J1 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)^7"), "")
presentation.add_rule(p, parse_relations("(ab(abaB)^3)^5"), "")
presentation.add_rule(p, parse_relations("(ab(abaB)^6abab(aB)^2)^2"), "")
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
Running the above script produces the following output:
Output from the Python script
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: RUN 0 START (strategy() = felsch)
#0: ToddCoxeter: |A| = 2, |R| = 9, |u| + |v| ∈ [2, 87], ∑(|u| + |v|) = 206
++++++++++++++++++++++++++++++++
#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 = 33µs | all runs = 33µs | elapsed = 99µs
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: FELSCH 0.0 STOP
#0: ToddCoxeter: FELSCH 0.0.1 | active | killed | defined
#0: ToddCoxeter: nodes | 175,560 | 263,717 | 439,277
#0: ToddCoxeter: diff 0.0.0 | +175,559 | +263,717 | +439,276
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 351,120 | 0 | 100.0%
#0: ToddCoxeter: diff 0.0.0 | +351,120 | -2 | +100.0%
#0: ToddCoxeter: phase 0.0 = 815ms | run 0 = 815ms | all runs = 815ms | elapsed = 815ms
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 START (lookahead_extent() = full, lookahead_style() = hlt)
#0: ToddCoxeter: LOOKAHEAD 0.1.0 | active | killed | defined
#0: ToddCoxeter: nodes | 175,560 | 263,717 | 439,277
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 351,120 | 0 | 100.0%
#0: ToddCoxeter: time | run 0 = 815ms | all runs = 815ms | elapsed = 815ms
#0: ToddCoxeter: triggered because there are skipped definitions (175,560 active nodes)!
++++++++++++++++++++++++++++++++
#0: ToddCoxeter: LOOKAHEAD 0.1 STOP
#0: ToddCoxeter: LOOKAHEAD 0.1.1 | active | killed | defined
#0: ToddCoxeter: nodes | 175,560 | 263,717 | 439,277
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0
#0: ToddCoxeter: | active | missing | % complete
#0: ToddCoxeter: edges | 351,120 | 0 | 100.0%
#0: ToddCoxeter: diff 0.1.0 | +0 | +0 | +0.0%
#0: ToddCoxeter: phase 0.1 = 111ms | run 0 = 926ms | all runs = 926ms | elapsed = 926ms
#0: ToddCoxeter: lookahead_next() is now max(f x a = 351,120, m = 10,000) (-4,648,880)
#0: ToddCoxeter: because f x a < n
#0: ToddCoxeter: where: a = number_of_nodes_active() = 175,560
#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 | 175,560 | 263,717 | 439,277
#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 | 351,120 | 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 = 111ms | run 0 = 926ms | all runs = 926ms | elapsed = 926ms
++++++++++++++++++++++++++++++++
#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 | 111ms (12%) | - (0%) | - (0%) | 815ms (88%)
#0: ToddCoxeter: phase 0.2 = 112ms | run 0 = 927ms | all runs = 927ms | elapsed = 927ms
The size of the group is 175560
The computed size of the group matches the size of the group provided on the ATLAS: \(175,560\).