Z5 in A5
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) cyclic group:Z5 and the group is (up to isomorphism) alternating group:A5 (see subgroup structure of alternating group:A5).
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The group is alternating group:A5, which for concreteness, we take as the alternating group on the set .
is the subgroup:
There is a total of five other conjugate subgroups (so six subgroups, including ). The other subgroups are:
|order of whole group||60|
|order of subgroup||5|
|index of a subgroup||12|
|size of conjugacy class of subgroup||6|
|number of conjugacy classes in automorphism class of subgroup||1|
|size of automorphism class of subgroup||6|
The group-subgroup pair can be constructed as follows:
G := AlternatingGroup(5); H := Group([(1,2,3,4,5)]);