Z5 in A5
From Groupprops
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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Definition
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:
Arithmetic functions
Function | Value | Explanation |
---|---|---|
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 |
GAP implementation
The group-subgroup pair can be constructed as follows:
G := AlternatingGroup(5); H := Group([(1,2,3,4,5)]);