A3 in A4

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) alternating group:A3 and the group is (up to isomorphism) alternating group:A4 (see subgroup structure of alternating group:A4).
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This article describes the subgroup $H$ in the group $G$. Here, $G$ is the alternating group:A4, acting on the set $\{ 1,2,3,4 \}$. $H$ is the subgroup:

$\! H = H_4 := \{ (), (1,2,3), (1,3,2) \}$

It has three other conjugates:

$\! H_1 := \{ (), (2,3,4), (2,4,3) \}, H_2 := \{ (), (1,3,4), (1,4,3) \}, H_3 := \{ (), (1,2,4), (1,4,2) \}$

With this notation, each $H_i$ is the stabilizer of $\{ i \}$ in $G$.

See also subgroup structure of alternating group:A4.

Cosets

Each of the four subgroups has four left cosets and four right cosets. Further, for every pair of subgroups, there is exactly one coset that is a left coset for the first subgroup and a right coset for the second subgroup.

Complements

All four subgroups have a unique common normal complement, which is the Klein four-subgroup of alternating group:A4:

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

This is also the unique permutable complement to each of them.

Also, each of $H_1, H_2, H_3, H_4$ has each of the following three subgroups as a lattice complement that is not a permutable complement:

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

Properties related to complementation

Property	Meaning	Satisfied?	Explanation	Comment
permutably complemented subgroup	has a permutable complement	Yes	See above
lattice-complemented subgroup	has a lattice complement	Yes	See above
retract	has a normal complement	Yes	See above
complemented normal subgroup	normal subgroup with permutable complement	No	Not normal

Arithmetic functions

Function	Value	Explanation
order of whole group	12
order of subgroup	3
index	4
size of conjugacy class	4
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	the subgroup itself	current page	cyclic group:Z3
centralizer	the subgroup itself	current page	cyclic group:Z3
normal core	trivial subgroup	--	trivial group
normal closure	whole group	--	alternating group:A4
characteristic core	trivial subgroup	--	trivial group
characteristic closure	whole group	--	alternating group:A4
commutator with whole group	subgroup $\{ (), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$	V4 in S4	Klein four-group

Related subgroups

Intermediate subgroups

The subgroup $H$ is a maximal subgroup of $G$, so there are no strictly intermediate subgroups between $H$ and $G$.

Smaller subgroups

The subgroup $H$ is a group of prime order, so it has no proper nontrivial subgroup.

Normality-related properties

For ease of reference, we take here the subgroup $H = \{ (), (1,2,3), (1,3,2) \}$, though the conclusions apply for the other three conjugates as well.

Some of the properties follow on account of being a Sylow subgroup.

Property	Meaning	Satisfied?	Explanation	Comment
normal subgroup	equals its conjugate subgroups	No
subnormal subgroup	has a chain to whole group each normal in next	No
pronormal subgroup		Yes
automorph-conjugate subgroup	all automorphic subgroups are conjugate subgroups	Yes
isomorph-conjugate subgroup		Yes
order-conjugate subgroup		Yes
order-automorphic subgroup		Yes
order-isomorphic subgroup		Yes