Symmetric group isomorphic to semidirect product of Alternating group and Z/2Z

Question

I'm having a hard time understanding why $A_n \rtimes \mathbb{Z}_2 \cong S_n$.

I understand that $A_n$ is normal in $S_n$. But that's about it. What would the $\alpha$: $\mathbb{Z}$$_2$$\longrightarrow$Aut($A_n$) be?

Answer 1

Pick an odd element of $ S_n $ of order two; for example, a transposition. Then conjugation by that element is an order two automorphism of $ A_n $.