A new proof for the Erd\H{o}s-Ko-Rado Theorem for the alternating group

A subset $S$ of the alternating group on $n$ points is {\it intersecting} if for any pair of permutations $\pi,\sigma$ in $S$, there is an element $i\in \{1,\dots,n\}$ such that $\pi(i)=\sigma(i)$. We prove that if $S$ is intersecting, then $|S|\leq \frac{(n-1)!}{2}$. Also, we prove that if $n \geq 5$, then the only sets $S$ that meet this bound are the cosets of the stabilizer of a point of $\{1,\dots,n\}$.