On almost fixed-point-free automorphisms of groups. (Q2795890)

It is well-known that any finite group \(G\), admitting a fixed-point-free automorphism of prime order, is nilpotent. An automorphism \(\alpha\) of an arbitrary group \(G\) is said to be \textit{almost fixed-point-free} if there are only finitely many elements of \(G\) fixed by \(\alpha\). In the paper under review, it is shown that if \(G\) is a finitely generated group such that the centre of its automorphism group contains an almost fixed-point-free automorphism, then the centre \(Z(G)\) has finite index in \(G\). Moreover, extending a previous result of \textit{G. Endimioni} [Arch. Math. 94, No. 1, 19-27 (2010; Zbl 1205.20041)], the author proves that if \(G\) is a nilpotent minimax residually finite group such that the Fitting subgroup of the automorphism group of \(G\) contains an almost fixed-point-free automorphism, then the commutator subgroup \(G'\) of \(G\) is periodic.