On the fixed-point set and commutator subgroup of an automorphism of a soluble group. (Q2248803)

Let \(G\) be a group, \(\varphi\in\Aut(G)\) and define the subgroups \(C_G(\varphi)=\{g\in G\mid g^\varphi=g\}\) and \([G,\varphi]=\langle g^{-1}g^\varphi\mid g\in G\rangle\). In Theorem 1 the author proves that if \(G\) is a nilpotent-by-abelian group of derived length \(d\), \(C_{G'}(\varphi)\) is a periodic \(\pi\)-group for a set \(\pi\) of primes and \(\varphi\) has prime order \(p\), then \([G,\varphi]G'\) is an extension of a \(\pi\)-group by a nilpotent group. Moreover \([G,\varphi]G'/O_\pi(G')\) is abelian if \(p=1\) and nilpotent of class at most \((p^d-1)/(p-1)\) if \(p\) is odd. In Theorem 2 it is proved that if \(G\) is metabelian, \(\varphi\) has prime order \(p\) and if \(B=C_{G'}(\varphi)^G\), then \([G,\varphi]C_G(\varphi)/B\) is nilpotent of class at most \(p\) (even \(1\) if \(p=2\)). Theorems 3 and 4 are devoted to the case in which the order of \(\varphi\) is \(2\) and \(C_G(\varphi)\) is finite or locally finite.