The classification of finite simple groups, whether it be viewed as finished, or as a work in progress, is (or will be) without doubt an enormous achievement. It clearly sheds a great deal of light on the structure of finite groups. However, as with the classification of simple Lie algebras, one might expect this to have a significant impact outside of the immediate subject. So what are some of the known, or expected, applications to the classification outside of finite group theory?
this question was significantly edited by other users soon after being posted (Aug 2 '10). The few critical comments below were actually addressed before the question was edited and improved.
To anyone who has doubts about the proof, please see Aschbacher's Notices paper "The status of the classification of the finite simple groups" and please rather don't debate that particular matter here.