Group embedding

A group embedding is an injective group homomorphism. It identifies its domain with an isomorphic subgroup of its codomain.

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 $(n+1n+2)$ 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.

Every group $G$ embeds in the symmetric group on its underlying set through the left-regular action $g:x\mapsto gx$.

For $n>1$, a cyclic group of prime-power order $n$ cannot embed in $S_{n-1}$, because a permutation of order $p^k$ requires a cycle of length at least $p^k$. If $n$ is not a prime power, every group of order $n$ embeds in $S_{n-1}$ by combining coset actions on subgroups of two distinct prime orders.