Cayley's Theorem - Imported Knowlpedia Content

Cayley's Theorem. Let $G$ be a group. Let $\operatorname{Sym}(G)$ denote the group of all bijections $\sigma: G \to G$ under composition. For each $g \in G$ , define the left translation map $L_g: G \to G$ by $L_g(x)=gx$ . Then the map

\lambda: G \to \operatorname{Sym}(G), \qquad \lambda(g)=L_g,

is an injective homomorphism (i.e. a monomorphism). Equivalently, $G$ is isomorphic to a subgroup of $\operatorname{Sym}(G)$ .

Remarks

Cayley's theorem says every abstract group can be realized concretely as a group of permutations. The construction comes from the left multiplication action of $G$ on itself, which is faithful and hence yields a permutation representation.