A3 in S4
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:Z3 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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This article is about the subgroup in the group
, where
is the symmetric group of degree four, acting on the set
, and
is the three-element subgroup:
In other words, is the alternating group on
viewed naturally as a subgroup of
.
There are three other conjugate subgroups of in
(so a total of four), with each conjugate characterized as the alternating group on some subset of size three. Indexing these by the fixed point, we get:
Contents
Complements
The permutable complements to (and also to each of its conjugates) are precisely the 2-Sylow subgroups of
, which are the D8 in S4s, namely, copies of dihedral group:D8 sitting inside the whole group. One such complement is:
There are also other lattice complements that are not permutable complements -- for instance, any Z4 in S4 is a lattice complement that is not a permutable complement.
Arithmetic functions
Function | Value | Explanation | Comment |
---|---|---|---|
order of whole group | 24 | ||
order of the subgroup | 3 | ||
index of the subgroup | 8 | ||
size of conjugacy class | 4 | ||
number of conjugacy classes in automorphism class | 1 |
Effect of subgroup operators
Function | Value as subgroup (descriptive) | Value as subgroup (link) | Value as group |
---|---|---|---|
normalizer | ![]() |
S3 in S4 | symmetric group:S3 |
centralizer | the subgroup itself | (current page) | cyclic group:Z3 |
normal core | trivial subgroup | -- | trivial group |
normal closure | ![]() |
A4 in S4 | alternating group:A4 |
characteristic core | trivial subgroup | -- | trivial group |
characteristic closure | ![]() |
A4 in S4 | alternating group:A4 |
Subgroup properties
The subgroup is a Sylow subgroup for the prime 3. Many properties follow from this fact.