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.
2026-03-28 03:00:26.1774666826
Proper subgroup of simple groups
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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?