TITLE: Order of an automorphism of a finite group
QUESTION [48 upvotes]: 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!)

REPLY [49 votes]: 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.