Every Finite Group is Isomorphic to a Subgroup of $A_n$ [closed]
How does one prove that every finite group is isomorphic to a subgroup of an alternating group?
By Cayley's theorem every group is isomorphic to a subgroup of $S_n $.
Then $S_n $ is isomorphic to a subgroup of $A_{n+2}$. To see this, map even permutations to themselves, and for odd permutations map to the even permutation gotten by multiplying by the transposition $(n+1 \ n+2) $. It is straightforward to check that this defines an embedding.
One can take the composition of the two embeddings: $G\hookrightarrow S_n\hookrightarrow A_{n+2}$, where $n=|G|$.