Cayley's theorem

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Statement

In terms of group actions

Let $G$ be a group. The group multiplication $G\times G\to G$, defines a group action of $G$ on itself. In other words, the left multiplication gives an action of $G$ on itself, with the rule $g.h=gh$.

Further, this group action is faithful -- no non-identity element of $G$ acts trivially.

This action is termed the left-regular group action.

In terms of homomorphisms

Let $G$ be a group. The action of $G$ on itself by left multiplication gives a homomorphism from $G$ to $\mathrm{S}\mathrm{y}\mathrm{m}(|G|)$ (the symmetric group, i.e., the group of all permutations, on the underlying set of $G$). Moreover, this homomorphism is injective. Thus, every group can be realized as a subgroup of a symmetric group.