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Author: Killian O'Brien
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A4 = AlternatingGroup(4) A4
Alternating group of order 4!/2 as a permutation group
A4.subgroups()
[Subgroup generated by [()] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(1,2)(3,4)] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(1,3)(2,4)] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(1,4)(2,3)] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(2,4,3)] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(1,2,3)] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(1,4,2)] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(1,3,4)] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(1,2)(3,4), (1,3)(2,4)] of (Alternating group of order 4!/2 as a permutation group), Subgroup generated by [(2,4,3), (1,2)(3,4), (1,3)(2,4)] of (Alternating group of order 4!/2 as a permutation group)]
[x for x in A4]
[(), (1,3)(2,4), (1,2)(3,4), (1,4)(2,3), (2,4,3), (1,3,4), (1,2,3), (1,4,2), (2,3,4), (1,3,2), (1,2,4), (1,4,3)]
[(x,x.order()) for x in A4]
[((), 1), ((1,3)(2,4), 2), ((1,2)(3,4), 2), ((1,4)(2,3), 2), ((2,4,3), 3), ((1,3,4), 3), ((1,2,3), 3), ((1,4,2), 3), ((2,3,4), 3), ((1,3,2), 3), ((1,2,4), 3), ((1,4,3), 3)]
y=A4("(1,3)(2,4)") y
(1,3)(2,4)
H = A4.subgroup([y]) H [x for x in H]
Subgroup generated by [(1,3)(2,4)] of (Alternating group of order 4!/2 as a permutation group) [(), (1,3)(2,4)]
[h for h in H]
[(), (1,2)(3,4)]