Elliptic curve n-torsion point?

For an elliptic curve $E/k$, let $\alpha$ be any endomorphism over $\bar{k}$ in End($E$) and let $[n]$ be the multiplication-by-n endomorphism. Show that for any n-torsion point $P \in E[n]$, we must have $\alpha(P) \in E[n]$.

I know and have just showed that $[n]$ is commutative with all endomorphisms but I'm not sure how to use that fact here. What's a good way to show $\alpha(P) \in E[n]$?

• In general, if two endomorphims commute, each stabilizes the kernel of the other. – Captain Lama Apr 12 '16 at 16:45
• @CaptainLama how exactly does that help here? – user3772119 Apr 12 '16 at 16:47
• Well, you say you showed that $[n]$ commutes with all $\alpha$ and you want to prove that $\alpha$ stabilizes the kernel of $[n]$. – Captain Lama Apr 12 '16 at 16:48