S2 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:Z2 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 pointwise. In particular, is the symmetric group on , embedded naturally in . It is isomorphic to cyclic group:Z2. As a set, contains two elements: and .
There are five other conjugate subgroups to in (so the total conjugacy class size of subgroups is 3). Each subgroup fixed pointwise a subset of size two. Equivalently, each subgroup comprises the identity element and a 2-transposition. Specifically, and its two other conjugate subgroups are:
See also subgroup structure of symmetric group:S4.
Each of these subgroups has 12 left cosets and 12 right cosets. Further, every left coset of one subgroup is a right coset of one of its conjugate subgroups. Overall, there are thus 72 cosets. Each coset is characterized by a fixed behavior on two of the four points.PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
There is a unique permutable complement to all these subgroups, which is also a normal subgroup and hence a normal complement. In particular, each of the subgroups is a retract. The unique permutable complement is A4 in S4, i.e., the alternating group of degree four.
|retract||has a normal complement||Yes||alternating group:A4 (see A4 in S4) is a complement.|
|permutably complemented subgroup||has a permutable complement||Yes||(via retract)|
|lattice-complemented subgroup||has a lattice complement||Yes||(via retract)|
|direct factor||normal subgroup with normal complement||No||not normal|
|complemented normal subgroup||normal subgroup with permutable complement||No||not normal|
|order of whole group||24|
|order of subgroup||2|
|index of the subgroup||12|
|size of conjugacy class||6|
|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||non-normal Klein four-subgroups of symmetric group:S4||Klein four-group|
|centralizer||non-normal Klein four-subgroups of symmetric group:S4||Klein four-group|
|normal core||trivial subgroup||trivial||trivial group|
|normal closure||whole group||--||symmetric group:S4|
|characteristic core||trivial subgroup||trivial||trivial group|
|characteristic closure||whole group||--||symmetric group:S4|
The values given here are specific to .
|Value of intermediate subgroup (descriptive)||Isomorphism class of intermediate subgroup||Number of conjugacy classes of intermediate subgroup fixing subgroup and whole group||Subgroup in intermediate subgroup||Intermediate subgroup in whole group|
|Klein four-group||1||Z2 in V4||non-normal Klein four-subgroups of symmetric group:S4|
|symmetric group:S3||3||S2 in S3||S3 in S4|
|dihedral group:D8||1||non-normal subgroups of dihedral group:D8||D8 in S4|
The subgroup has order two, hence is minimal, and has no smaller nontrivial subgroups.
|normal subgroup||equals all its conjugate subgroups||No||See list of conjugate subgroups|
|2-subnormal subgroup||normal in its normal closure||No||Normal closure is whole group|
|subnormal subgroup||there is a series from the subgroup to the whole group, each normal in the next||No|
|contranormal subgroup||normal closure is the whole group||Yes||Normal closure is whole group||transpositions generate the finitary symmetric group|
|self-normalizing subgroup||equals its normalizer in the whole group||No||Normalizer is|
|hypernormalized subgroup||taking normalizer repeatedly reaches the whole group||No||Normalizer is self-normalizing|
|pronormal subgroup||any conjugate subgroup to the subgroup is conjugate to it in their join||No||The subgroup is a conjugate, but in their join , which is abelian, the two subgroups are not conjugates.|
|weakly pronormal subgroup||No|
|weakly normal subgroup||No|
|order-isomorphic subgroup||any subgroup of the whole group of the same order is isomorphic to it||Yes||any subgroup of order two is isomorphic to cyclic group:Z2|
|isomorph-automorphic subgroup||any subgroup of the whole group isomorphic to it is related to it by an automorphism||No||The subgroup is isomorphic but there is no automorphism sending to this subgroup.||This other subgroup is subgroup generated by double transposition in symmetric group:S4.|
|automorph-conjugate subgroup||any subgroup automorphic to it is conjugate to it||Yes||The whole group is a complete group -- all automorphisms of it are inner automorphisms.||See symmetric groups are complete|