Not sure how to do this:
Fix integer n>1. Prove there exist only finitely many simple groups containing proper subgroups of index smaller than or equal to n.
Not sure how to do this:
Fix integer n>1. Prove there exist only finitely many simple groups containing proper subgroups of index smaller than or equal to n.
If $G$ is a simple group containing a subgroup $H$ of index $m \le n$, then the action on cosets gives a homomorphism $G \to S_m$. What can the kernel of this homomorphism be?