S4 in S5
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) symmetric group:S4 and the group is (up to isomorphism) symmetric group:S5 (see subgroup structure of symmetric group:S5).
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Contents
Definition
The group is taken as symmetric group:S5: the symmetric group of degree five. For concreteness, we take
as the symmetric group on the set
.
We take as the subgroup fixing
, so
is symmetric group:S4 acting on the set
.
has four other conjugate subgroups, each corresponding to a different fixed point:
-
is the subgroup fixing
, and is the symmetric group on the set
.
-
is the subgroup fixing
, and is the symmetric group on the set
.
-
is the subgroup fixing
, and is the symmetric group on the set
.
-
is the subgroup fixing
, and is the symmetric group on the set
.
-
is the subgroup fixing
, and is the symmetric group on the set
.
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 | ![]() |
|
p-complement | complement of a ![]() |
Yes | ![]() ![]() |
GAP implementation
The group-subgroup pair can be constructed as follows:
G := SymmetricGroup(5); H := SymmetricGroup(4);