Order of an automorphism of a finite group

Question

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!)

Nope, I didn't know the answer. But now I do! – David E Speyer Oct 19 '09 at 00:36 — David E Speyer, Oct 19 '09 at 00:36

score 49 · Accepted Answer · answered Oct 18 '09 at 21:24

49

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.

answered Oct 18 '09 at 21:24

Greg Stevenson

8,541

1

Great! Now if only I could read Russian... :) – Reid Barton Oct 18 '09 at 22:20
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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
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This was fun to read--very clever, entirely elementary arguments--thanks again. – Reid Barton Oct 19 '09 at 07:22
No worries, glad I could help. – Greg Stevenson Oct 19 '09 at 08:28
This is currently Problem 11681 in AMM Vol. 119, No. 10 (December 2012). – Benjamin Dickman Nov 21 '12 at 15:05
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The proof is given in the book of Isaacs on Finite Group Theory (p.70). – Philip Oct 29 '15 at 04:00

Order of an automorphism of a finite group

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