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

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 $p^{\mathrm{\prime }}$-group admitting a splitting automorphism of prime order is locally nilpotent if $$⟨g,g^{\phi },\dots ,g^{{\phi }^{p-1}}⟩$$ 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.