Symmetric Group on 4 Letters/Subgroups/Examples/Disjoint Transpositions

Example of Subgroup of Symmetric Group on 4 Letters
Let $H$ be the subset of the Symmetric Group on $4$ Letters $S_4$ which consists of the $3$ products of disjoint transpositions of $S_4$, and the identity:


 * $H := \set {e, \tuple {1 2} \tuple {3 4}, \tuple {1 3} \tuple {2 4}, \tuple {1 4} \tuple {2 3} }$

Then $H$ forms a subgroup of $S_4$.

The Cayley table of $H$ can be presented as:


 * $\begin{array}{c|cccc}

\circ                    &                         e & \tuple {1 2} \tuple {3 4} & \tuple {1 3} \tuple {2 4} & \tuple {1 4} \tuple {2 3} \\ \hline e &                        e & \tuple {1 2} \tuple {3 4} & \tuple {1 3} \tuple {2 4} & \tuple {1 4} \tuple {2 3} \\ \tuple {1 2} \tuple {3 4} & \tuple {1 2} \tuple {3 4} &                        e & \tuple {1 4} \tuple {2 3} & \tuple {1 3} \tuple {2 4} \\ \tuple {1 3} \tuple {2 4} & \tuple {1 3} \tuple {2 4} & \tuple {1 4} \tuple {2 3} &                        e & \tuple {1 2} \tuple {3 4} \\ \tuple {1 4} \tuple {2 3} & \tuple {1 4} \tuple {2 3} & \tuple {1 3} \tuple {2 4} & \tuple {1 2} \tuple {3 4}                        & e \\ \end{array}$

This is the Klein $4$-group.

$H$ is normal in $S_4$.

The quotient group $S_4 / H$ is the Symmetric Group on $3$ Letters $S_3$.

Proof
We have that $H$ contains all the elements of $S_4$ of the same cycle type.

From Cycle Decomposition of Conjugate, the conjugate of a permutation is another permutation of the same cycle type.

Hence the conjugate of an element of $H$ is an element of $H$.

That is, $H$ is normal in $S_4$.

The quotient group $S_4 / H$ is of order $\dfrac {24} 4 = 6$, so must be either $C_6$ or $S_3$.

It remains to be shown that $S_4 / H$ is non-abelian.