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

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 $$p'$$ p ′ -group admitting a splitting automorphism of prime order is locally nilpotent if $$ \langle g, g^\varphi , \dots , g^{\varphi ^{p-1}} \rangle $$ ⟨ g , g φ , ⋯ , g φ p - 1 ⟩ is nilpotent for every $$g \in G$$ g ∈ G [7, Problem 10.59]. We prove that if $$G$$ G is a periodic residually finite $$p'$$ p ′ -group admitting a splitting automorphism of prime order $$p,$$ p , then $$G$$ G is nilpotent of class bounded in terms of $$p$$ 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.