A3 in A4



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:

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

Effect of subgroup operators
In the table below, we provide values specific to $$H$$.

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 satisfies property::Sylow subgroup.