MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!)

share|cite|improve this question
Nope, I didn't know the answer. But now I do! – David Speyer Oct 19 '09 at 0:36
up vote 43 down vote accepted

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

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.