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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.

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Finite intersection perserves finite index. Consider the order of the overgroup in relation to the index of the normal core of this proper subgroup in view of simplicity. – anon Nov 3 '12 at 3:53
I've changed algebra tag to abstract-algebra, since we don't use algebra tag anymore, see meta for details. – Martin Sleziak Nov 3 '12 at 6:05

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?

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