Non-normal Klein four-subgroups of symmetric group: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) Klein four-group 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 Young subgroup for the partition . Explicitly, it is the subgroup comprising those permutations that send each of the subsets and to within itself. has a total of three conjugates, listed below:
|Our local name||Partition stabilized||Set of all elements in the stabilizing subgroup|
(and hence each of its conjugate subgroups) is isomorphic to the Klein four-group. However, has another subgroup isomorphic to the Klein four-group that is not one of these conjugate subgroups, and is not automorphic to these either. That is the normal Klein four-subgroup of symmetric group:S4 that comprises the identity element and the three double transpositions. The current article is not about that subgroup.
There is a total of 18 cosets, each of which is a left coset for exactly one subgroup and a right coset for exactly one subgroup. Moreover, each coset is parametrized by the way it sends one partition (labeled) to another. Table below is incomplete.
|Subset in cycle decomposition notation||Subset in one-line notation||Source and target subsets of||Left coset of ?||Left coset of ?||Left coset of ?||Right coset of ?||Right coset of ?||Right coset of ?|
|1234, 2134, 1243, 2143||,||Yes||No||No||Yes||No||No|
|1234, 3214, 1432, 3412||,||No||Yes||No||No||Yes||No|
|1234, 4231, 1324, 4321||,||No||No||Yes||No||No||Yes|
|3421, 4312, 3412, 4321||,||Yes||No||No||Yes||No||No|
|2341, 4123, 2143, 4321||,||No||Yes||No||No||Yes||No|
|2413, 3142, 2143, 3412||,||No||No||Yes||No||No||Yes|
None of these subgroups has a permutable complement. All the subgroups do have a common lattice complement; in fact, there is a conjugacy class of subgroups each of which is conjugate to each of these subgroups. That conjugacy class is the conjugacy class of A3 in S4, which includes each of these subgroups:
|retract||has a normal complement||No|
|permutably complemented subgroup||has a permutable complement||No|
|lattice-complemented subgroup||has a lattice complement||Yes|
|direct factor||normal subgroup with normal complement||No|
|complemented normal subgroup||normal subgroup with permutable complement||No|
|order of whole group||24|
|order of subgroup||4|
|index of the subgroup||6|
|size of conjugacy class||3|
|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||D8 in S4||dihedral group:D8|
|centralizer||the subgroup itself||(current page)||Klein four-group|
|normal core||trivial subgroup||--||trivial group|
|normal closure||the whole group||--||symmetric group:S4|
|characteristic core||trivial subgroup||--||trivial group|
|characteristic closure||the whole group||--||symmetric group:S4|
|normal subgroup||equals all its conjugate subgroups||No||(see above for other conjugate subgroups)|
|2-subnormal subgroup||normal subgroup in its normal closure||No|
|subnormal subgroup||series from subgroup to whole group, each normal in next||No|
|contranormal subgroup||normal closure is whole group||Yes|
|self-normalizing subgroup||equals normalizer in the whole group||No||normalizer is dihedral group of order eight|
|self-centralizing subgroup||contains its centralizer in the whole group||Yes|
|subgroup whose join with any distinct conjugate is the whole group||join of the subgroup with any distinct conjugate subgroup is the whole group||Yes|
|pronormal subgroup||any conjugate to it is conjugate in their join||Yes|
|weakly pronormal subgroup||Yes|
|order-isomorphic subgroup||isomorphic to every subgroup of the group of the same order||No||The subgroup has the same order but is isomorphic to cyclic group:Z4|
|isomorph-automorphic subgroup||any subgroup of the group isomorphic to it is automorphic to it||No||There is also the subgroup||See normal Klein four-subgroup of symmetric group:S4|