Finite groups admitting a dihedral group of automorphisms

Abstract

Let D=⟨α,β⟩ be a dihedral group generated by the involutions α and β and let F=⟨αβ⟩. Suppose that D acts on a finite group G by automorphisms in such a way that CG(F)=1. In the present paper we prove that the nilpotent length of the group G is equal to the maximum of the nilpotent lengths of the subgroups CG(α) and CG(β).