Symmetric group S_4 - Knowlpedia

The symmetric group $S_4$ is the group of all bijections of $\{1,\ldots,4\}$ , with composition as multiplication.

Order and normal subgroup

There are 24 permutations: choose the images of the 4 letters successively. The even permutations form the proper nontrivial normal subgroup $A_4$ , the kernel of the sign map. Therefore $S_4$ is not simple.

Klein subgroup

The identity and the three double transpositions form a normal Klein four-group. Conjugation permutes these three nonidentity elements.

References

J. S. Milne, Group Theory, v4.01, §4, symmetric and alternating groups; Theorem 4.33 and Remark 4.34, p. 69.