S3 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) symmetric group:S3 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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We consider the subgroup in the group
defined as follows.
is the symmetric group of degree four, which, for concreteness, we take as the symmetric group on the set
.
is the subgroup of
comprising those permutations that fix
. In particular,
is the symmetric group on
, embedded naturally in
. It is isomorphic to symmetric group:S3.
has order
.
There are three other conjugate subgroups to in
(so the total conjugacy class size of subgroups is 4). The other subgroups are the subgroups fixing
,
, and
respectively.
The four conjugates are:
See also subgroup structure of symmetric group:S4.
Cosets
There are four left cosets and four right cosets of each subgroup. Each left coset of a subgroup is a right coset of one of its conjugate subgroups. This gives a total of 16 subsets.
The cosets are parametrized by ordered pairs . The coset parametrized by
is the set of all elements that send
to
. This is a left coset of
and a right coset of
.
Complements
There is a unique normal complement that is common to all the subgroups. This is the subgroup normal Klein four-subgroup of symmetric group:S4:
There is also a conjugacy class of subgroups each of which is a permutable complement to each of the s. These are cyclic four-subgroups of symmetric group:S4, and these are:
Note that the fact that these are permutable complements can be understood as a special case of Cayley's theorem. See also every group of given order is a permutable complement for symmetric groups, which says that any finite group of order is, via the Cayley embedding, a permutable complement to
in
.
Apart from these, each of the s has a number of lattice complements:
- Any subgroup generated by double transposition in S4 is a lattice complement to each
in the whole group. Thus, each
has three such lattice complements.
- For each
, a subgroup of order three not contained in that
is a lattice complement to it. Thus, each
has three such lattice complements, because one of the four subgroups of order three is contained in that
.
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
retract | has a normal complement | Yes | subgroup ![]() |
|
permutably complemented subgroup | has a permutable complement | Yes | normal complement is permutable complement too | |
lattice-complemented subgroup | has a lattice complement | Yes | normal complement is lattice complement too | |
complemented normal subgroup | normal subgroup with permutable complement | No | not normal itself |
Arithmetic functions
Function | Value | Explanation |
---|---|---|
order of whole group | 24 | |
order of subgroup | 6 | |
index of subgroup | 4 | |
size of conjugacy class of subgroup (=index of normalizer) | 4 | see above for list of conjugates |
number of conjugacy classes in automorphism class of subgroup | 1 | the whole group is a complete group, so the conjugation actions are precisely the automorphisms. |
size of automorphism class of subgroup | 4 | same as size of conjugacy class |