Automorphisms of finite order of nilpotent groups IV

Wehrfritz, B.A.F.

Rendiconti del Seminario Matematico della Università di Padova, Tome 136 (2016), pp. 61-68.

Let $\varphi$ be an automorphism of finite order of the nilpotent group $G$ of class $c$ and $m$ and $r$ positive integers with ${\varphi }^m=1$. Consider the two (not usually homomorphic) maps $\psi$ and $\gamma$ of $G$ given by

$$\psi :g⟼g·g\varphi ·g{\varphi }^2·...·g{\varphi }^{m-1}\text{and}\gamma :g\mapsto g^{-1}·g\varphi \text{for}g\in G.$$

We prove that the subgroups

$$X=\langle x\alpha :x\in ker\psi ,\alpha \in \mathrm{Aut}G,x^r\in {\bigcup }_{s\ge 0}{\left(G\gamma \right)}^s\rangle ,$$

$$Y=\langle g\gamma \alpha :g\in G,\alpha \in \mathrm{Aut}G,{\left(g\gamma \right)}^r\in \ker \gamma \rangle ,$$

$$X^{*}=\langle x^r\alpha :x\in \ker \psi ,\in \alpha \in \mathrm{Aut}G,x^r\in {\bigcup }_{s\ge 0}{\left(G\psi \right)}^s\rangle ,$$

$$Y^{*}=\langle {\left(g\gamma \right)}^r\alpha :g\in G,\alpha \in \mathrm{Aut}G,{\left(g\gamma \right)}^r\in \ker \gamma \rangle =\langle ({\left(G\gamma \right)}^r\cap \ker \gamma )\mathrm{Aut}G\rangle$$

of $G$ all have finite exponent bounded in terms of $c$, $m$ and $r$ only. This yields alternative proofs of the theorem of [4] and its related bounds.