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:
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.
|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|
The subgroup is a Sylow subgroup for the prime 3. Many properties follow from this fact.