On the centralizer and the commutator subgroup of an automorphism. (Q1759664)

Let \(\varphi\) be an automorphism of a group \(G\). It is proved that if \(G\) is polycyclic, \(|\varphi|=2\), and \(C_G(\varphi )\) is finite, then \([G,\varphi]'\) is finite. It is also proved that if \(G\) is polycyclic and \(C_G(\varphi)\) is finite, then \(G/[G,\varphi]\) is finite (for any \(\varphi\)). If in addition \(|\varphi|\) is finite, then \(|G:C_G(\varphi)[G,\varphi]|\) is finite. There are also results on interrelation between \(C_G(\varphi)\) and \([G,\varphi]\) for metabelian groups with automorphisms of finite order, and for soluble groups with an automorphism of order 2.