# Two groups that are the automorphism groups of each other

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 ?

• 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. – Mark Bennet Jan 8 '15 at 20:51
• Can there be a cycle with more than one element? – Jorge Fernández Hidalgo Jan 8 '15 at 20:54
• – Jorge Fernández Hidalgo Jan 8 '15 at 20:56
• 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. – Nicky Hekster Jan 8 '15 at 20:59
• @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. – Derek Holt Jan 10 '15 at 14:44