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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Definition

The group $G$ is alternating group:A5, which for concreteness, we take as the alternating group on the set $\{1,2,3,4,5\}$.

$H$ is the subgroup:

$\!H:=\{(),(1,2,3,4,5),(1,3,5,2,4),(1,4,2,5,3),(1,5,4,3,2)\}$

There is a total of five other conjugate subgroups (so six subgroups, including $H$). The other subgroups are:

• $\!\{(),(1,2,3,5,4),(1,3,4,2,5),(1,5,2,4,3),(1,4,5,3,2)\}$
• $\!\{(),(1,2,4,3,5),(1,4,5,2,3),(1,3,2,5,4),(1,5,3,4,2)\}$
• $\!\{(),(1,2,4,5,3),(1,4,3,2,5),(1,5,2,3,4),(1,3,5,4,2)\}$
• $\!\{(),(1,2,5,3,4),(1,5,4,2,3),(1,3,2,4,5),(1,4,3,5,2)\}$
• $\!\{(),(1,2,5,4,3),(1,5,3,2,4),(1,4,2,3,5),(1,3,4,5,2)\}$

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)]);