Cayley's Theorem and Abelian Groups

Question

Is Cayley's theorem applicable to Abelian groups? It states that every group is isomorphic to a subgroup of a permutation group. However permutation groups are not Abelian.

Thanks

Answer 1

Yes, Cayley's theorem is applicable to any finite group, including abelian groups. The action of left multiplication by an element $g\in G$ is a permutation of $G$, and different elements give different permutations, so that we can view $G$ inside of $S_{|G|}$.

Note: subgroups of non-abelian groups can themselves be abelian.

Answer 2

Yes. Full permutation groups $S_{n}$ are not Abelian for $n>2$, but subgroup may be. For example permutation subgroup $G<S_{4}$ (with cycle convention) $$ G=\{e, (12), (34), (12)(34)\} $$