A3 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:Z3 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
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This article is about the subgroup H in the group G, where G is the symmetric group of degree four, acting on the set \{ 1,2,3,4 \}, and H is the three-element subgroup:

\! H =\{ (), (1,2,3), (1,3,2) \}

In other words, H is the alternating group on \{ 1,2,3 \} viewed naturally as a subgroup of G.

There are three other conjugate subgroups of H in G (so a total of four), with each conjugate characterized as the alternating group on some subset of size three. Indexing these by the fixed point, we get:

H_1 = \{ (), (2,3,4), (2,4,3 \}, \qquad H_2 = \{ (), (1,3,4), (1,4,3) \}, \qquad H_3 = \{ (), (1,2,4), (1,4,2) \}, H_4 = H = \{ (), (1,2,3), (1,3,2) \}

Complements

The permutable complements to H (and also to each of its conjugates) are precisely the 2-Sylow subgroups of G, which are the D8 in S4s, namely, copies of dihedral group:D8 sitting inside the whole group. One such complement is:

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

There are also other lattice complements that are not permutable complements -- for instance, any Z4 in S4 is a lattice complement that is not a permutable complement.

Properties related to complementation

Property Meaning Satisfied? Explanation Related examples
permutably complemented subgroup has a permutable complement Yes Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
lattice-complemented subgroup has a lattice complement Yes Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
retract has a normal complement No Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property

Arithmetic functions

Function Value Explanation Comment
order of whole group 24
order of the subgroup 3
index of the subgroup 8
size of conjugacy class 4
number of conjugacy classes in automorphism class 1

Effect of subgroup operators

Function Value as subgroup (descriptive) Value as subgroup (link) Value as group
normalizer \{ (), (1,2), (2,3), (1,3), (1,2,3), (1,3,2) \} S3 in S4 symmetric group:S3
centralizer the subgroup itself (current page) cyclic group:Z3
normal core trivial subgroup -- trivial group
normal closure \langle (1,2)(3,4), (1,2,3) \rangle A4 in S4 alternating group:A4
characteristic core trivial subgroup -- trivial group
characteristic closure \langle (1,2)(3,4), (1,2,3) \rangle A4 in S4 alternating group:A4

Subgroup properties

The subgroup is a Sylow subgroup for the prime 3. Many properties follow from this fact.

Normality-related properties

Property Meaning Satisfied? Explanation Related examples
normal subgroup equals all its conjugate subgroups No (see other conjugate subgroups) Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
contranormal subgroup normal closure is whole group No normal closure is A4 in S4 Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
self-normalizing subgroup equals its normalizer in whole group No normalizer is S3 in S4 Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
pronormal subgroup any conjugate is conjugate in their join Yes Sylow implies pronormal Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property

Resemblance-based properties

Property Meaning Satisfied? Explanation Related examples
order-conjugate subgroup conjugate to any subgroup of the whole group with the same order Yes Sylow implies order-conjugate Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
order-dominating subgroup any subgroup whose order divides the order of H is contained in a conjugate of H Yes Sylow implies order-dominating Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
order-dominated subgroup any subgroup whose order is a multiple of the order of H contains a conjugate of H Yes Sylow implies order-dominated Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
order-isomorphic subgroup isomorphic to any subgroup of the whole group of the same order Yes (via order-conjugate), also because there is a unique isomorphism class of groups of order three Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
isomorph-automorphic subgroup automorphic to any subgroup of the whole group that is isomorphic to it Yes (via order-conjugate) Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
automorph-conjugate subgroup conjugate to any subgroup of the whole group that is isomorphic to it Yes (via order-conjugate), also because symmetric group:S4 is a complete group (see symmetric groups are complete) so every automorphism is inner Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
order-automorphic subgroup Yes (via order-conjugate) Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property
isomorph-conjugate subgroup Yes (via order-conjugate) Find all subgroups of the same big group satisfying the property | Find all subgroups of the same big group dissatisfying the property| Find all occurrences of the subgroup in other big groups satisfying the property | Find all subgroups of the same big group dissatisfying the property