I came accross a beautiful and "simple" (no pun intended) theorem, mentioned here in slide 14 by Jack Schmidt.
All finite simple groups have a cyclic Sylow $p$-subgroup for some $p$
I found references to proofs that involve the classification of finite simple groups, for instance in Composition factors from the group ring and Artin's theorem on orders of simple groups by Wolfgang Kimmerle , Richard Lyons , Robert Sandling , David N. Teague (theorem 4.9).
Is there a known proof that does not involve the classification of finite simple groups? That would be really nice.