How to prove in three ways that $A_n$ is a normal subgroup of $S_n$.

Question

I know how to prove in one way that $A_n$ is a subgroup of $S_n$ but I do not know how to prove that $A_n$ is a normal subgroup of $S_n$ in three different way

Answer 1

Here's an outline of the methods I would use:

1. Argue using the fact that $[S_n: A_n]=2$. (You can either use the cosets directly, or else argue that the normalizer of $A_n$ has to be strictly larger than $A_n$.)
2. Argue using the fact that the parity of transpositions of $h$ and $ghg^{-1}$ are the same.
3. Show that $A_n$ is the kernel of a group homomorphism of $S_n\to \{-1, 1\}$.