$S_n$ embeds into $A_{n+2}$.

Question

I was reading this post explaining why $S_n$ embeds into $A_{n+2}$. So they suggest that the embedding schould be
$Sn \hookrightarrow A_{n+2}$
$\sigma \mapsto \sigma$ if $\sigma$ is even and
$\sigma \mapsto \sigma (n+1, n+2)$ if $\sigma$ is odd. Could I multiply by any transposition in the case where $\sigma$ is odd or is it important to take $(n+1, n+2)$?

Answer 1

If you multiply by another transposition the function might not be a homomorphism. The reason why it works with $(n+1,n+2)$ is because this specific transposition commutes with any permutation $\sigma\in S_n$.