Subgroup generated by double transposition in 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) 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 generated by the double transposition . This is the permutation that interchanges with and with . Since the element has order two, is a two-element subgroup, isomorphic to cyclic group:Z2, and its two elements are the identity and .
There are two other conjugate subgroups to in (so the total conjugacy class size of subgroups is 3). The three subgroups are given below:
See also subgroup structure of symmetric group:S4.
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(and hence also each of its conjugate subgroups) has no permutable complements. However, it does have a lattice complement. Specifically, any S3 in S4 is a lattice complement to and also to each of its conjugates. For instance, .
|retract||has a normal complement||No|
|permutably complemented subgroup||has a permutable complement||No|
|lattice-complemented subgroup||has a lattice complement||Yes|
|normal subgroup||equals all its conjugate subgroups||No|
|2-subnormal subgroup||normal subgroup of normal subgroup||Yes||Normal in , which is the normal Klein four-subgroup of symmetric group:S4|
|hypernormalized subgroup||taking normalizers repeatedly reaches the whole group||No||Normalizer is D8 in S4, which is self-normalizing|
|pronormal subgroup||any conjugate is conjugate to it in their join||No||pronormal and subnormal implies normal|
|contranormal subgroup||normal closure is whole group||No|
|self-normalizing subgroup||equals its normalizer in the whole group||No|
|order-isomorphic subgroup||isomorphic to any subgroup of the same order||Yes||Any group of order two is isomorphic to cyclic group:Z2|
|isomorph-automorphic subgroup||automorphic to any subgroup isomorphic to it||No||The subgroup is isomorphic to it but not related to it via an automorphism||See S2 in S4 for this other automorphism class.|
|automorph-conjugate subgroup||conjugate to any subgroup automorphic to it||Yes|