S4 in S5

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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) symmetric group:S4 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
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Definition

The group G is taken as symmetric group:S5: the symmetric group of degree five. For concreteness, we take G as the symmetric group on the set \{ 1,2,3,4,5 \}.

We take H = H_5 as the subgroup fixing \{ 5 \}, so H is symmetric group:S4 acting on the set \{ 1,2,3,4 \}.

H has four other conjugate subgroups, each corresponding to a different fixed point:

  • H_1 is the subgroup fixing \{ 1 \}, and is the symmetric group on the set \{ 2,3,4,5 \}.
  • H_2 is the subgroup fixing \{ 2 \}, and is the symmetric group on the set \{ 1,3,4,5 \}.
  • H_3 is the subgroup fixing \{ 3 \}, and is the symmetric group on the set \{ 1,2,4,5 \}.
  • H_4 is the subgroup fixing \{ 4 \}, and is the symmetric group on the set \{ 1,2,3,5 \}.
  • H = H_5 is the subgroup fixing \{ 5 \}, and is the symmetric group on the set \{ 1,2,3,4 \}.

Arithmetic functions

Function Value Explanation
order of group 120
order of subgroup 24
index of subgroup 5
size of conjugacy class of subgroup 5
number of conjugacy classes in automorphism class of subgroup 1
size of automorphism class of subgroup 5

Subgroup properties

Other properties

Property Meaning Satisfied? Explanation Comment
Hall subgroup order and index are relatively prime Yes \{ 2,3 \}-Hall subgroup. Also, order (24) and index (5) are relatively prime.
p-complement complement of a p-Sylow subgroup Yes p-complement for p = 5

GAP implementation

The group-subgroup pair can be constructed as follows:

G := SymmetricGroup(5); H := SymmetricGroup(4);