Finite groups as permutation and even-permutation groups

For each \(g\in G\), let \[ L_g:G\longrightarrow G, \qquad L_g(x)=gx. \] Because multiplication by \(g\) has inverse multiplication by \(g^{-1}\), each \(L_g\) is a permutation of the underlying set \(G\). Moreover, \[ L_g\circ L_h=L_{gh}, \] so \[ \lambda:G\longrightarrow \operatorname{Sym}(G), \qquad \lambda(g)=L_g \] is a homomorphism. If \(L_g\) is the identity permutation, then \[ g=L_g(e)=e, \] so \(\lambda\) is injective. Hence \[ G\cong \lambda(G)\le \operatorname{Sym}(G)\cong S_n. \] This is Cayley’s theorem.

By step 1 it suffices to embed \(S_n\) into an alternating group. Regard \(S_n\) as the permutations of \(\{1,\ldots,n+2\}\) that fix \(n+1\) and \(n+2\), and let \[ \tau=(n+1\ n+2). \] Then \(\tau\) commutes with every element of this copy of \(S_n\).

Define \[ \Phi:S_n\longrightarrow A_{n+2} \] by \[ \Phi(\sigma)= \begin{cases} \sigma,&\sigma\text{ even},\\ \sigma\tau,&\sigma\text{ odd}. \end{cases} \] Equivalently, if \(\varepsilon(\sigma)\in\{0,1\}\) is the parity of \(\sigma\), then \[ \Phi(\sigma)=\sigma\tau^{\varepsilon(\sigma)}. \] The permutation \(\Phi(\sigma)\) is always even, because an odd \(\sigma\) is multiplied by the odd transposition \(\tau\).

Since \(\tau\) commutes with \(S_n\) and parity is additive modulo \(2\), \[ \begin{aligned} \Phi(\sigma\rho) &=\sigma\rho\,\tau^{\varepsilon(\sigma\rho)}\\ &=\sigma\rho\,\tau^{\varepsilon(\sigma)+\varepsilon(\rho)}\\ &=\sigma\tau^{\varepsilon(\sigma)}\rho\tau^{\varepsilon(\rho)}\\ &=\Phi(\sigma)\Phi(\rho). \end{aligned} \] Thus \(\Phi\) is a homomorphism. If \(\Phi(\sigma)=e\), then restricting to the first \(n\) letters gives \(\sigma=e\), so \(\Phi\) is injective. Therefore \[ S_n\hookrightarrow A_{n+2}. \] Composing this embedding with the Cayley embedding of \(G\) proves that \(G\) is isomorphic to a subgroup of an alternating group, hence to a group of even permutations.

Step 1 proves part 1, and step 2 proves part 2.