Suppose $G$ is a finite simple group in which every proper subgroup is abelian. If $M$ and $N$ are distinct maximal subgroups of $G$ show that $M \cap N = 1$.
My plan for this problem is to use abelianess of proper subgroups of $G$ to produce a map out of $G$ with kernel $M \cap N$. I am not sure if I am on the right track.