A5 in S5
This article is about a particular subgroup in a group, up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (up to isomorphism) alternating group:A5 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
The subgroup is a normal subgroup and the quotient group is isomorphic to cyclic group:Z2.
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We define as symmetric group:S5 -- for concreteness, the symmetric group on the set .
is alternating group:A5 -- the subgroup of comprising even permutations. A criterion for a permutation to be even, based on cycle decomposition, is that the number of cycles of even length should be even.
|order of group||120||Order is|
|order of subgroup||60||Order is|
|index of subgroup||2||For , the alternating group has index two in the symmetric group, since it has two cosets: itself (the even permutations) and the odd permutations.|
|Subgroup-defining function||Meaning in general||Why it takes this value||GAP verification (set G := SymmetricGroup(5); H := AlternatingGroup(5);)|
|derived subgroup (also called commutator subgroup)||subgroup generated by commutators||Since the quotient is abelian, it contains the derived subgroup. Further, is simple non-abelian, so the derived subgroup cannot be smaller.||H = DerivedSubgroup(G); using DerivedSubgroup|
|hypocenter||stable member of lower central series (transfinite if necessary)||is the derived subgroup, and|
|perfect core||stable member of derived series (transfinite if necessary); alternatively, join of all perfect subgroups||is perfect, and the only bigger subgroup is , which is not perfect because|
G := SymmetricGroup(5); H := AlternatingGroup(5);