A subgroup is normal iff it is invariant under every inner automorphism. How to prove that An⊴Sn.

Question

Well i know how to prove using the index method but i have no idea of proving this by using the above condition.

Answer 1

$A_n$ is released as the kernel of the signature on $S_n$, whence is normal.

Another way to see it, $A_n$ has index $2$ in $S_n$.

Answer 2

Hint:

For any $\;\pi\in A_n\,,\,\,\,\sigma\in S_n\;$ , what is the sign of $\;\sigma^{-1}\pi\sigma\;$ ?