Status of the automorphism tower problem for finite groups

Asked May 14 '23 at 02:51

Active May 14 '23 at 11:00

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This is problem 11.123 in the Kourovka notebook:

For a given group $G$, define the following sequence of groups: $A_1(G) = G$, $A_{i+1}(G) = \operatorname{Aut}(A_i(G))$. Does there exist a finite group $G$ for which this sequence contains infinitely many non-isomorphic groups? (M. Short)

I'm familiar with Wielandt's negative answer in the special case that $\mathbf{Z}(G)=1$. Have any other advances been made?

edited May 14 '23 at 11:00

LSpice

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asked May 14 '23 at 02:51

semisimpleton

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Maybe https://sites.math.rutgers.edu/~sthomas/tower.pdf is helpful. – Benjamin Steinberg May 14 '23 at 03:21
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Just to have it linked, although it's not an answer: Does $\operatorname{Aut}(\operatorname{Aut}(\cdots\operatorname{Aut}(G)\cdots))$ stabilize?. – LSpice May 14 '23 at 11:02

Status of the automorphism tower problem for finite groups

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