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.
Contents
Cosets
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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,
.
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
Property | Meaning | Satisfied? | Explanation | Comment |
---|---|---|---|---|
normal subgroup | equals all its conjugate subgroups | No | ||
2-subnormal subgroup | normal subgroup of normal subgroup | Yes | Normal in ![]() |
|
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 ![]() |
See S2 in S4 for this other automorphism class. |
automorph-conjugate subgroup | conjugate to any subgroup automorphic to it | Yes |