Finite groups admitting a dihedral group of automorphisms

Abstract

Let \(D=\langle \alpha, \beta \rangle\) be a dihedral group generated by the involutions \(\alpha\) and \(\beta\) and let \(F=\langle \alpha \beta \rangle\). Suppose that \(D\) acts on a finite group \(G\) by automorphisms in such a way that \(C_G(F)=1\). 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 \(C_G(\alpha)\) and \(C_G(\beta)\).