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Almost fixed-point-free automorphisms of soluble groups. (English) Zbl 1215.20030
An automorphism \(\varphi\) of a group \(G\) is said to be almost-regular if the set \(C_G(\varphi)\) is finite. G. Endimioni, [in J. Algebra 323, No. 11, 3142-3146 (2010; Zbl 1202.20037)], proved that a polycyclic-by-finite group \(G\) with an almost-regular automorphism of prime order is nilpotent-by-finite and that a polycyclic-by-finite group \(G\) with an almost-regular automorphism of order \(4\) contains a characteristic subgroup \(H\) of finite index such that \(H''\leq Z(H)\).
The aim of the paper under review is to show that Endimioni’s results will remain valid for wide classes of groups. In particular the author proves that, in order to obtain the same conclusions, it is sufficient to assume one of the following hypotheses: (i) is \(G\) a finite extension of a torsion free soluble minimax group or (ii) \(G\) a soluble-by-finite subgroup of \(\mathrm{GL}(n,F)\), where \(n\in\mathbb N\) and \(F\) is a finite extension field of \(\mathbb Q\).

MSC:
20E36 Automorphisms of infinite groups
20F16 Solvable groups, supersolvable groups
20F19 Generalizations of solvable and nilpotent groups
20F28 Automorphism groups of groups
20D45 Automorphisms of abstract finite groups
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