Consequences of the Classification of Finite Simple Groups

Question

There are many conjectures in finite group theory which were solved thanks to the Classification theorem. Here are some of the most famous ones ($S$ will always be a non-abelian finite simple group):

1. Schreier's conjecture: $\operatorname{Out}(S)$ is solvable.
2. Ore's conjecture: every element of $S$ is a commutator.
3. The only 4- and 5-fold transitive permutation groups are the symetric alternating, and Mathieu groups.

Do you know any other examples of this?

Answer 1

Nikolov and Segal have proved important conjectures on profinite groups using the classification of finite simple groups. A survey here is given by B. Klopsch, see this paper. Jean-Pierre Serre had asked a problem on finite abstract quotients of profinite groups, in the context of Galois cohomology. Nikolov and Segal answered this and proved a variety of important results.