Automorphisms of finite order of nilpotent groups. III. (Q496363)

Let \(\varphi\) be an automorphism of order \(m\) of the group \(G\). One can define the maps \[ \psi\colon G\to G,\quad g\mapsto g\cdot g^\varphi\cdots g^{\varphi^{m-1}},\qquad\gamma\colon G\to G,\quad g\mapsto g^{-1}g^\varphi \] and the sets \(\ker\eta=\{g\in G\mid g^\eta=1\}\), \(G^\eta=\{g^\eta\mid g\in G\}\) (\(\eta\in\{\psi,\gamma\}\)) of which only \(\ker\gamma=C_G(\varphi)\) is always a subgroup of \(G\). In a previous paper of this series [part II, Stud. Sci. Math. Hung. 51, No. 4, 547-555 (2014; Zbl 1331.20043)] the author has shown that if \(G\) is a nilpotent FAR-group, then \(G^\psi\cdot\ker\psi\) and \(G^\gamma\cdot\ker\gamma\) are both large in the sense that they contain subgroups of finite index in \(G\). In the paper under review the author proves that, under the same hypotheses, \(\langle G^\psi\cap\ker\psi\rangle\) and \(\langle G^\gamma\cap\ker\gamma\rangle\) are both finite subgroups. Thus \(\psi\) and \(\gamma\) determine two weak splittings of \(G\) as product of subsets.