Z5 in A5

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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)]);