Automorphisms of finite order of Nilpotent groups

Abstract

Let $$\phi $$ be an automorphism of prime order $$p$$ of the nilpotent group $$G$$ . Frequently one needs to consider the map $$\psi : G\rightarrow G$$ given by $$\begin{aligned} g\psi =g\cdot g\phi \cdot g\phi ^2\cdot \ldots \cdot g\phi ^{p-1}. \end{aligned}$$ This $$\psi $$ is not usually a homomorphism of $$G$$ and $$ker\psi $$ and $$G\psi $$ need not be subgroups. Under relatively mild restrictions we prove that $$G\psi $$ is finite if and only if $$ker\psi $$ contains a subgroup of $$G$$ of finite index and $$ker\psi $$ is finite if and only if $$G\psi $$ contains a subgroup of finite index. These results do not extend to polycyclic groups or Cernikov groups.