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!)



Order of an automorphism of a finite group
Yes every automorphism has order bounded by |G|-1, provided G is not the trivial group. A reference is  

M V Horoševskiĭ 1974 Math. USSR Sb. 22 584-594  

which can be found at  

<http://www.iop.org/EJ/abstract/0025-5734/22/4/A08/>


It is even shown that the upper bound is reached only for elementary abelian groups.


