A residually finite analogue of Kegel's theorem on splitting automorphisms

TL;DR

This paper proves that periodic residually finite groups with a splitting automorphism of prime order are nilpotent with bounded class, affirmatively answering a problem posed by Sozutov.

Contribution

It establishes that such groups are necessarily nilpotent of bounded class, extending classical results to residually finite groups and ruling out certain counterexamples.

Findings

Residually finite groups with a splitting automorphism of prime order are nilpotent.

The nilpotency class is bounded in terms of the prime order.

Counterexamples to Sozutov's problem cannot be Tarski monsters.

Abstract

Thompson proved that every finite group admitting a fixed-point-free automorphism of prime order is nilpotent, and Kegel showed that the same conclusion holds for finite groups admitting a splitting automorphism of prime order. Motivated by these results, Sozutov asked whether a \(p'\)-group admitting a splitting automorphism of prime order is locally nilpotent if \[ \langle g, g^\varphi, \dots, g^{\varphi^{p-1}} \rangle \] is nilpotent for every \(g \in G\), \cite[Problem 10.59]{kourovka21}. We prove that if \(G\) is a periodic residually finite group admitting a splitting automorphism of prime order \(p\) then \(G\) is nilpotent of class bounded in terms of \(p\). This gives an affirmative answer, for residually finite groups, to the problem of Sozutov. We also prove that a possible counterexample to Sozutov's problem cannot be a Tarski monster.