Non-coprime automorphisms of finite groups

We present some recent results about non-coprime automorphisms of finite groups and discuss several open problems.

I. Finite groups admitting an automorphism with fixed-point subgroup of given order or rank. There are strong results about solubility or/and bounds for the Fitting height of such groups in the case of coprime automorphism. But in the case of non-coprime automorphism some important problems remain open. The main new result is a bound in terms of $m$ and $n$ for the Fitting height of a finite soluble group admitting an automorphism of order $n$ with $m$ fixed points. As a corollary, Hartley's problem about locally finite groups is solved in the affirmative.

II. Local--global generation property of commutators. (The results discussed in part II are joint work with C. Acciarri, R. M. Guralnick, and P. Shumyatsky.) Let $A$ be a group of automorphisms of a group $G$. An application of Schur's theorem yields that if the set of commutators $\{[g,a]\mid g\in G,\; a\in A\}$ is finite of cardinality $m$, then the commutator subgroup $[G,A]$ is finite of order bounded above in terms of $m$. This observation is generalized for finite groups by replacing cardinality with a certain natural rank parameter, with the aim of bounding the Pr"ufer rank of $[G,A]$. A result reported at this seminar during my visit to ICM-Shenzhen in 2025 about a group $A$ of coprime automorphisms is now proved for non-coprime automorphisms under the unavoidable $\pi(A)$-solubility condition.