Alternating-group embedding of a finite group
= Alternating-group embedding of a finite group
{title2=$G\hookrightarrow A_{|G|+2}$}
<Cayley theorem> embeds a <finite group> of order $n$ into $S_n$. Extend $\sigma\in S_n$ to $n+2$ letters and multiply it by the disjoint <transposition> $(n+1\ n+2)$ exactly when $\sigma$ is odd. Additivity of parity makes this an injective <group homomorphism> into the <alternating group> $A_{n+2}$. Thus every <finite group> is isomorphic to a <subgroup> of some <alternating group>.