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).
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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