Subgroup generated by double transposition in symmetric group:S4

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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 H in the group G defined as follows.

G is the symmetric group of degree four, which, for concreteness, we take as the symmetric group on the set \{ 1,2,3 \}.

H is the subgroup of G generated by the double transposition (1,2)(3,4). This is the permutation that interchanges 1 with 2 and 3 with 4. Since the element has order two, H is a two-element subgroup, isomorphic to cyclic group:Z2, and its two elements are the identity and (1,2)(3,4).

There are two other conjugate subgroups to H in G (so the total conjugacy class size of subgroups is 3). The three subgroups are given below:

H = \{ (), (1,2)(3,4) \}, \qquad H_1 = \{ (), (1,4)(2,3) \}, \qquad H_2 = \{ (), (1,3)(2,4) \}

See also subgroup structure of symmetric group:S4.

Cosets

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Complements

H (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 H and also to each of its conjugates. For instance, \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \}.

Properties related to complementation

Property Meaning Satisfied? Explanation Comment
retract has a normal complement No
permutably complemented subgroup has a permutable complement No
lattice-complemented subgroup has a lattice complement Yes

Subgroup properties

Normality-related properties

Property Meaning Satisfied? Explanation Comment
normal subgroup equals all its conjugate subgroups No
2-subnormal subgroup normal subgroup of normal subgroup Yes Normal in \{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}, 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

Resemblance-based properties

Property Meaning Satisfied? Explanation Comment
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 \{ (), (1,2) \} 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