A note on groups admitting a uniform automorphism of prime power order (Q755888)

An automorphism \(\alpha\) of a group G is called uniform if every element of G has the form \(x^{-1}x^{\alpha}\) for some x in G. It is well- known that an automorphism of a finite group is uniform if and only if it is fixed-point-free. It was proved by \textit{D. J. S. Robinson} [Invent. Math. 10, 38-43 (1970; Zbl 0198.345)] that if a finitely generated hyperabelian group G has a uniform automorphism of prime order p, then G is a finite nilpotent group with order prime to p. It is shown in this paper that if G is a finitely generated hyperabelian group admitting a uniform automorphism of order \(p^ n\) (p prime), then G is a finite group with order prime to p. This result was already proved in a previous article of the same author [Group Theory, Proc. 2nd Int. Conf., Bressanone/Italy 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 23, 197-199 (1990; Zbl 0699.20028)].