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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    $\begingroup$ Nope, I didn't know the answer. But now I do! $\endgroup$ – David E 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

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

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    $\begingroup$ Great! Now if only I could read Russian... :) $\endgroup$ – Reid Barton Oct 18 '09 at 22:20
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    $\begingroup$ The link is to an english translation ;) $\endgroup$ – Greg Stevenson Oct 18 '09 at 22:40
  • $\begingroup$ So it is! (I googled for a freely available copy and found one in Russian...) $\endgroup$ – Reid Barton Oct 18 '09 at 23:06
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    $\begingroup$ This was fun to read--very clever, entirely elementary arguments--thanks again. $\endgroup$ – Reid Barton Oct 19 '09 at 7:22
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    $\begingroup$ The proof is given in the book of Isaacs on Finite Group Theory (p.70). $\endgroup$ – Philip Oct 29 '15 at 4:00

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