Klein four-group - Knowlpedia

The Klein four-group $V_4$ is the direct product $C_2\times C_2$ , with coordinatewise addition modulo two. It has four elements, and every nonidentity element has order two.

Permutation realization

The subgroup $\{1,(12)(34),(13)(24),(14)(23)\}$ of $A_4$ is isomorphic to $V_4$ . Multiplying any two distinct double transpositions gives the third. Conjugation preserves cycle type, so this subgroup is normal in both $A_4$ and $S_4$ .

Nonsimplicity

Each nonidentity element generates a subgroup of order two. The group is abelian, so these subgroups are normal; it is not simple.

References

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