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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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    I've changed [tag:algebra] tag to [tag:abstract-algebra], since we don't use algebra tag anymore, see [meta](http://meta.math.stackexchange.com/questions/473/the-use-of-the-algebra-tag/3081#3081) for details.2012-11-03

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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?