Let $H,K$ be two non-isomorphic groups such that $H\cong Aut(K)$ and $K\cong Aut(H)$.

Is there any example of such groups ?

Note: I had asked the question there.

  • $\begingroup$ I think this question is answered in the comments to this question: mathoverflow.net/questions/5635/… $\endgroup$
    – HJRW
    Jan 10, 2015 at 19:06
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    $\begingroup$ @HJRW: I could not find which comment ? $\endgroup$
    – mesel
    Jan 10, 2015 at 19:10
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    $\begingroup$ I couldn't find the answer there either, although it seems hard to believe that nobody has thought about this problem before, or indeed whether there are any known examples of longer finite cycles of automorphism groups. $\endgroup$
    – Derek Holt
    Jan 10, 2015 at 23:01
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    $\begingroup$ If the initial group is finite and centerless then there are no finite cycles of length more than $1$. $\endgroup$
    – Pablo
    Jan 10, 2015 at 23:04
  • $\begingroup$ Sorry, I was too hasty. I meant for the finite case with no centre. $\endgroup$
    – HJRW
    Jan 11, 2015 at 19:00

1 Answer 1


Computer search in sage/gap didn't found any solutions for orders up to $120$.

It didn't assume the orders are equal.


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