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 ?

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    $\begingroup$ I'm wondering, and may ask separately, what behaviour is known for taking the Automorphism Group multiple times for finite groups - what are the fixed points (Symmetric Groups other than $S_6$ are included)? Are non-trivial cycles known? Can the groups become unboundedly large, or must every chain end in a fixed point or a cycle? Interesting question, which has got me thinking. $\endgroup$ Jan 8, 2015 at 20:51
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    $\begingroup$ Can there be a cycle with more than one element? $\endgroup$
    – Asinomás
    Jan 8, 2015 at 20:54
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    $\begingroup$ related mathoverflow.net/questions/151352/iterated-automorphism-groups $\endgroup$
    – Asinomás
    Jan 8, 2015 at 20:56
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    $\begingroup$ Wielandt [1939] proved the classical result that the automorphism tower of any centerless finite group terminates in finitely many steps. Rae and Roseblade [1970] proved that the automorphism tower of any centerless Cernikov group terminates in finitely many steps. Hulse [1970] proved that the the automorphism tower of any centerless polycyclic group terminates in countably many steps. Simon Thomas [1985] proved that the automorphism tower of any centerless group eventually terminates. Hamkins proved that every group has a terminating transfinite automorphism tower. Centerless or not. $\endgroup$ Jan 8, 2015 at 20:59
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    $\begingroup$ @mesel As far as I can see, Hamkin's results do not imply that there is no nontrivial cycle of automorphism groups of finite groups. He would just consider the direct limit of the whole sequence, and then start again from there. It seems strange that there appears to ne reference to this particular problem in the literature, even as an unsolved problem. It would be worth asking it on MO. $\endgroup$
    – Derek Holt
    Jan 10, 2015 at 14:44

1 Answer 1


An answer in the negative for this question would immediately prove the Smith conjecture: Does the finite part of the automorphism tower of a group G eventually repeat? (Smith 1964)

I have spent some time wrestling with this conjecture, and the pattern I observed was that this did occur and typically very rapidly (only a few steps in, I was using GAP to do the computations, as order of the groups also typically grew rapidly), moreover, every time I found a group for which it began repeating, it was also a reptend of length 1 (meaning the 'last' group was isomorphic to its automorphism group).

If the Smith conjecture is true, and reptend is always length 1, then your question's answer is false, otherwise if the Smith conjecture is true, so is your conjecture, but if Smith's conjecture is false, then it's unclear whether or not you conjecture is true or false.


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