Cayley's Theorem

From AoPSWiki

Cayley's Theorem states that every group is isomorphic to a permutation group, i.e., a subgroup of a symmetric group; in other words, every group acts faithfully on some set. Although the result is simple, it is deep, as it characterizes group structure as the structure of a family of bijections.

Proof

We prove that each group $G$ is isomorphic to a group of bijections on itself. Indeed, for all $g\in G$ , let $f_g$ be the mapping $f_g : x \mapsto gx$ from $G$ into itself. Then $f_g$ is a bijection, for all $g$ ; and for all $g,h \in G$ , $f_g \circ f_h = f_{gh}$ . Thus $G$ is isomorphic to the set of permutations $\{ f_g | g \in G\}$ on $G$ . $\blacksquare$