S2 in S5

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:S2 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 $\{1,2,3,4,5\}$.

Then for any order two element $\sigma \in G$, $\langle \sigma \rangle$ is a copy of symmetric group:S2 within symmetric group:S5.

Arithmetic functions

Function	Value	Explanation
order of group	120
order of subgroup	2
index of subgroup	60

Other properties

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup		No

GAP implementation

The group-subgroup pair can be constructed as follows:

G := SymmetricGroup(5); H := SymmetricGroup(2);