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 '15 at 19:06
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    $\begingroup$ @HJRW: I could not find which comment ? $\endgroup$ – mesel Jan 10 '15 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 '15 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 '15 at 23:04
  • $\begingroup$ Sorry, I was too hasty. I meant for the finite case with no centre. $\endgroup$ – HJRW Jan 11 '15 at 19:00

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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