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

Thompson proved that every finite group admitting a fixedpoint-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 ⟨g,gϕ,...,gϕ^(p−1)⟩ is nilpotent for every g E G [7, Problem 10.59]. We prove that if G is a periodic residually finite p′-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 ounterexample to Sozutov’s problem cannot be a Tarski monster.