S2 in 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 \{ 1,2,3,4 \}.

H is the subgroup of G comprising those permutations that fix \{ 3,4 \} pointwise. In particular, H is the symmetric group on \{ 1, 2\}, embedded naturally in G. It is isomorphic to cyclic group:Z2. As a set, H contains two elements: () and (1,2).

There are five other conjugate subgroups to H in G (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, H and its two other conjugate subgroups are:

\{ (), (1,2) \}, \qquad \{ (), (2,3) \}, \qquad \{ (), (3,4) \}, \qquad \{ (), (1,3) \}, \qquad \{ (), (2,4) \}, \qquad \{ (), (1,4) \}

See also subgroup structure of symmetric group:S4.

Cosets

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.

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Complements

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.

Properties related to complementation

Property Meaning Satisfied? Explanation Comment
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

Arithmetic functions

Function Value Explanation
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 H.

Function Value as subgroup (descriptive) Value as subgroup (link) Value as group
normalizer \{ (), (1,2), (3,4), (1,2)(3,4) \} non-normal Klein four-subgroups of symmetric group:S4 Klein four-group
centralizer \{ (), (1,2), (3,4), (1,2)(3,4) \} 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

Related subgroups

Intermediate subgroups

The values given here are specific to H.

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
\{ (), (1,2), (3,4), (1,2)(3,4) \} Klein four-group 1 Z2 in V4 non-normal Klein four-subgroups of symmetric group:S4
\{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \} symmetric group:S3 3 S2 in S3 S3 in S4
\{ (), (1,2), (3,4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (1,3,2,4), (1,4,2,3) \} dihedral group:D8 1 non-normal subgroups of dihedral group:D8 D8 in S4

Smaller subgroups

The subgroup has order two, hence is minimal, and has no smaller nontrivial subgroups.

Subgroup properties

Normality-related properties

Property Meaning Satisfied? Explanation Comment
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 \{ (), (1,2), (3,4), (1,2)(3,4) \}
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 \{ (), (3,4) \} is a conjugate, but in their join \{ (), (1,2), (3,4), (1,2)(3,4) \}, which is abelian, the two subgroups are not conjugates.
weakly pronormal subgroup No
paranormal subgroup No
polynormal subgroup No
weakly normal subgroup No

Resemblance-based properties

Property Meaning Satisfied? Explanation Comment
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 \{ (), (1,2)(3,4) \} is isomorphic but there is no automorphism sending H 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