Let G be a finite group of order n. Must every automorphism of G have order less than n?
(David Speyer: I got this question from you long ago, but I don't know whether you knew the answer. I stil don't!)
Yes every automorphism has order bounded by |G|-1, provided G is not the trivial group. A reference is
It is even shown that the upper bound is reached only for elementary abelian groups.