# A3 in 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:Z3 and the group is (up to isomorphism) symmetric group:S4 (see subgroup structure of symmetric group:S4).
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part | All pages on particular subgroups in groups

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.

## 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.

### 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