# Order of an automorphism of a finite group

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!)

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Nope, I didn't know the answer. But now I do! –  David Speyer Oct 19 '09 at 0:36

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.

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Great! Now if only I could read Russian... :) –  Reid Barton Oct 18 '09 at 22:20
The link is to an english translation ;) –  Greg Stevenson Oct 18 '09 at 22:40
So it is! (I googled for a freely available copy and found one in Russian...) –  Reid Barton Oct 18 '09 at 23:06
This was fun to read--very clever, entirely elementary arguments--thanks again. –  Reid Barton Oct 19 '09 at 7:22
No worries, glad I could help. –  Greg Stevenson Oct 19 '09 at 8:28
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