A3 in A4
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) alternating group:A3 and the group is (up to isomorphism) alternating group:A4 (see subgroup structure of alternating group:A4).
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This article describes the subgroup in the group . Here, is the alternating group:A4, acting on the set . is the subgroup:
It has three other conjugates:
With this notation, each is the stabilizer of in .
Each of the four subgroups has four left cosets and four right cosets. Further, for every pair of subgroups, there is exactly one coset that is a left coset for the first subgroup and a right coset for the second subgroup.
This is also the unique permutable complement to each of them.
Also, each of has each of the following three subgroups as a lattice complement that is not a permutable complement:
|permutably complemented subgroup||has a permutable complement||Yes||See above|
|lattice-complemented subgroup||has a lattice complement||Yes||See above|
|retract||has a normal complement||Yes||See above|
|complemented normal subgroup||normal subgroup with permutable complement||No||Not normal|
|order of whole group||12|
|order of subgroup||3|
|size of conjugacy class||4|
|number of conjugacy classes in automorphism class||1|
Effect of subgroup operators
In the table below, we provide values specific to .
|Function||Value as subgroup (descriptive)||Value as subgroup (link)||Value as group|
|normalizer||the subgroup itself||current page||cyclic group:Z3|
|centralizer||the subgroup itself||current page||cyclic group:Z3|
|normal core||trivial subgroup||--||trivial group|
|normal closure||whole group||--||alternating group:A4|
|characteristic core||trivial subgroup||--||trivial group|
|characteristic closure||whole group||--||alternating group:A4|
|commutator with whole group||subgroup||V4 in S4||Klein four-group|
The subgroup is a maximal subgroup of , so there are no strictly intermediate subgroups between and .
The subgroup is a group of prime order, so it has no proper nontrivial subgroup.
For ease of reference, we take here the subgroup , though the conclusions apply for the other three conjugates as well.
Some of the properties follow on account of being a Sylow subgroup.
|normal subgroup||equals its conjugate subgroups||No|
|subnormal subgroup||has a chain to whole group each normal in next||No|
|automorph-conjugate subgroup||all automorphic subgroups are conjugate subgroups||Yes|