Embedding $S_n$ in $A_{2n}$

Question

I want to embed the symmetric group $S_n$ into the bigger alternating group $A_{2n}$. How could I find such an injective homomorphism?

Answer 1

Proposition Let $n\ge 2$. The symmetric group $S_n$ can be embedded into the alternating group $A_k$ if and only if $k\ge n+2$.

A proof can be found here, and also at the link Jyrki has given in the comments. So $S_n$ cannot be embedded into $A_{n+1}$, for $n>1$, but $S_n\hookrightarrow A_{n+2}\hookrightarrow A_{n+3}\hookrightarrow A_{n+4}\hookrightarrow \cdots $, etc. is fine.