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?

  • 3
    $\begingroup$ 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. $\endgroup$ – Derek Holt Oct 29 '17 at 15:44
  • $\begingroup$ 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). $\endgroup$ – YCor Oct 29 '17 at 22:52

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.


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