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 .
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.
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 .
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 .
|retract||has a normal complement||Yes||subgroup above is a normal complement|
|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|
|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|