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On Groups of Automorphisms of Nilpotent p-Groups of Finite Rank

Let α and β be automorphisms of a nilpotent p-group G of finite rank. Suppose that 〈(αβ(g))(βα(g))–1: g ∈G〉 is a finite cyclic subgroup of G, then, exclusively, one of the following statements holds for G and Γ, where Γ is the group generated by α and β. G is finite, then Γ is an extension of a p-group by an abelian group. G is infinite, then Γ is soluble and abelian-by-finite. G is finite, then Γ is an extension of a p-group by an abelian group. G is infinite, then Γ is soluble and abelian-by-finite.