45
$\begingroup$

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!)

$\endgroup$
  • 7
    $\begingroup$ Nope, I didn't know the answer. But now I do! $\endgroup$ – David E Speyer Oct 19 '09 at 0:36
49
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Great! Now if only I could read Russian... :) $\endgroup$ – Reid Barton Oct 18 '09 at 22:20
  • 1
    $\begingroup$ 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
  • 1
    $\begingroup$ This was fun to read--very clever, entirely elementary arguments--thanks again. $\endgroup$ – Reid Barton Oct 19 '09 at 7:22
  • 3
    $\begingroup$ The proof is given in the book of Isaacs on Finite Group Theory (p.70). $\endgroup$ – Philip Oct 29 '15 at 4:00

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

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