Hello,

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.

Thank you.

knownproof without the classification. And such a proof seems impossible for mere mortals to achieve. – Jim Humphreys Oct 14 '12 at 19:57