Klein four-subgroup of alternating group:A4

This article describes the subgroup $$H$$ in the group $$G$$. Here, $$G$$ is the alternating group of degree four, acting on the set $$\{ 1,2,3,4 \}$$ for concreteness. In other words, $$G$$ is the set:

$$\! G := \{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3), (1,2,3), (1,3,2), (1,3,4), (1,4,3), (2,3,4), (2,4,3), (1,2,4), (1,4,2) \}$$

The subgroup $$H$$ is defined as the subgroup comprising the identity element and the three double transpositions, which can be characterized as the only even permutations that have no fixed points:

$$\! H := \{, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$$

The subgroup is isomorphic to the Klein four-group. It is a normal subgroup and the quotient group is isomorphic to cyclic group:Z3.

See also subgroup structure of alternating group:A4.



Cosets
$$H$$ is a normal subgroup of $$G$$, hence each left coset is a right coset and vice versa. The three cosets are then:

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

Some interesting observations:


 * Each of the cosets other than the subgroup itself is itself a conjugacy class. This makes the alternating group a Camina group.
 * Each 3-cycle and its inverse are contained in distinct cosets of $$H$$. This makes sense, if we note that the cyclic subgroup generated by the 3-cycle must be a permutable complement to $$H$$ in $$G$$.

Complements
$$H$$ is a satisfies property::complemented normal subgroup in $$G$$. There are four distinct possibilities for the complement of $$H$$ in $$G$$, all of which are conjugate subgroups. This also follows from the Schur-Zassenhaus theorem, since $$H$$ is a satisfies property::normal Sylow subgroup of $$G$$.

The four complements are:

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

For information on these as subgroups, see A3 in A4. Each of these is isomorphic to cyclic group:Z3, since complement to normal subgroup is isomorphic to quotient.

Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as the result of many subgroup-defining functions. Some of these are given below.

Intermediate subgroups
The subgroup has prime index, hence is maximal, so there are no strictly intermediate subgroups between the subgroup and the whole group.

Smaller subgroups
There are three proper nontrivial subgroups of the subgroup, all of which are conjugate inside the whole group (but not within the subgroup, which is abelian). The three subgroups are:

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

Each of these is isomorphic to cyclic group:Z2. For information on these as subgroups inside $$H$$, see Z2 in V4. For information on these as subgroups inside $$G$$, see subgroup generated by double transposition in A4.

Images under quotient maps
Under any quotient map with a nontrivial kernel, the image of the subgroup is trivial. This is because the group is a monolithic group and the subgroup is the unique minimal normal subgroup in it.

Invariance under automorphisms and endomorphisms: properties
Unless otherwise specified, each of these properties follows on account of the subgroup being a normal Sylow subgroup and in particular a normal Hall subgroup.