Normal Klein four-subgroup of symmetric group:S4

This article discusses the normal subgroup in the symmetric group of degree four comrpising the identity and the three double transpositions.

We let $$G = S_4$$ be the symmetric group of degree four, acting on $$\{ 1,2,3,4 \}$$ and $$H$$ be the subgroup of $$G$$ given by:

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

Cosets
The six cosets of this subgroup are as follows:

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

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

Complements
There are four possible complements to $$H$$ in $$G$$, all of which are conjugate subgroups and all are isomorphic to symmetric group:S3. These are the stabilizers of the individual points $$1,2,3,4$$:

Each of these is isomorphic to the quotient group $$G/H$$, since complement to normal subgroup is isomorphic to quotient.

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

Subgroup-defining functions
The subgroup is a characteristic subgroup of the whole group and arises as the result of many common subgroup-defining functions on the whole group.

Interpretation in terms of Cayley's theorem
We can think of the embedding of $$H$$ in $$G$$ in terms of Cayley's theorem. Specifically, think of starting with $$H$$ as an abstract Klein four-group whose elements are labeled $$1,2,3,4$$. Left multiplication by elements of $$H$$ induces precisely the identity and the three double transpositions on $$H$$ as a set. This thus makes $$H$$ a subgroup of the symmetric group on $$\{ 1,2,3,4 \}$$, which is $$G$$.

Note that the other, non-normal Klein four-subgroup cannot be interpreted this way because its non-identity elements are not fixed-point-free. However, the cyclic four-subgroup of the symmetric group of degree four can be embedded in this way.

Interpretation in terms of holomorph
We can think of the embedding of $$H$$ in $$G$$ as the abstract group $$H$$ sitting inside its holomorph. This is because $$G$$ is the semidirect product of $$H$$ and its automorphism group, which is $$GL(2,2)$$, which is isomorphic to the symmetric group of degree three.