Hölder’s quotient groups: S_4 / V_4 is S_3

Finds the normal Klein four-group V_4 inside S_4, forms the quotient S_4/V_4, and shows that it is a non-abelian group of order 6, like S_3.

from mathematicskit.abstract_algebra import PermutationGroup, all_subgroups, is_normal_subgroup, quotient_group
from mathematicskit.abstract_algebra.visualizers.plots import plot_cayley_table

Normal subgroups of S_4

s4 = PermutationGroup(4)
normal = [h for h in all_subgroups(s4) if is_normal_subgroup(s4, h)]
print(f"normal subgroup orders in S_4: {[len(h) for h in normal]}")

normal subgroup orders in S_4: [1, 4, 12, 24]

The quotient by the Klein four-group

v4 = next(h for h in normal if len(h) == 4)
q = quotient_group(s4, v4)
print(f"|S_4 / V_4| = {q.order}, abelian: {q.is_abelian()}")
print(f"element orders in the quotient: {sorted(q.element_order(c) for c in q.elements)} (same as S_3)")
plot_cayley_table(q)

|S_4 / V_4| = 6, abelian: False
element orders in the quotient: [1, 2, 2, 2, 3, 3] (same as S_3)

<Axes: title={'center': 'Cayley table (order 6)'}, xlabel='element index', ylabel='element index'>