Cayley's Theorem

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

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

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 .

Proof sketch. The rule $g \mapsto L_g$ respects multiplication because $L_{gh} = L_g \circ L_h$. Injectivity follows from $L_g = \mathrm{id}$ implying $g = L_g(e)=e$.