# Consequences of the Classification of Finite Simple Groups

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: $\operatorname{Aut}(S)$ 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?

• It doesn't make sense to call the odd-order theorem a corollary of the classification, because the odd-order theorem is used in the proof of the classification. – Derek Holt Oct 29 '17 at 15:44
• 1) Note that unlike 1 and 2 which are statement of the form "every finite simple group satisfies [something]", 3 is not a statement about finite simple groups. 2) the solution of Ore's conjecture is far from a direct consequence (it was proved 30 years after the classification was first announced). – YCor Oct 29 '17 at 22:52