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$\begingroup$

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](http://groupprops.subwiki.org/wiki/Index_satisfies_intersection_inequality). Consider the order of the overgroup in relation to the index of the [normal core](http://groupprops.subwiki.org/wiki/Normal_core) of this proper subgroup in view of simplicity.2012-11-03
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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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