Past exam of the mathematics course of the University of Cambridge / 2013 / ia / Paper 3 / 5D / a / Solution

Take $X=G$ and let $g$ act by left multiplication $L_g(x)=gx$. Each $L_g$ is a permutation, with inverse $L_{g^{-1}}$. Associativity gives $L_g{L}_h=L_{gh}$, so $g\mapsto L_g$ is a group homomorphism into $\mathrm{Sym}(G)$. If $L_g$ is the identity permutation, evaluating it at the identity element gives $g=1$. ThusThis is the Cayley theorem, realized by the left regular action, which is a faithful group action.