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Solvable groups equipped with an automorphism of prime order with finite centralizer. (Italian. English summary) Zbl 1247.20043

An automorphism \(\alpha\) of a group \(G\) is said to be ‘regular’ if \(C_G(\alpha)=\{1\}\), while \(\alpha\) is ‘quasi-regular’ if the centralizer \(C_G(\alpha)\) is finite. It has been proved by G. Endimioni [J. Algebra 323, No. 11, 3142-3146 (2010; Zbl 1202.20037)] that if \(G\) is a polycyclic group admitting a quasi-regular automorphism of prime order, then \(G\) contains a nilpotent subgroup of finite index. On the other hand, the same author [G. Endimioni, Arch. Math. 94, No. 1, 19-27 (2010; Zbl 1205.20041)] exhibited a regular automorphism of order 2 of the wreath product of two infinite cyclic groups, so that the above result cannot be extended to the case of finitely generated soluble groups.
In the paper under review, the authors prove that if \(G\) is any finitely generated soluble group with finite torsion-free rank admitting a quasi-regular automorphism of prime order \(p\), then \(G\) contains a nilpotent subgroup of finite index whose nilpotency class is bounded by a function of \(p\). Recall here that a group has ‘finite torsion-free rank’ if it has a series of finite length whose factors either are periodic or infinite cyclic. A corresponding result is proved in the case of minimax soluble group. As a special case, it follows that if \(G\) is a minimax soluble group admitting a quasi-regular automorphism of order 2, then \(G\) contains an Abelian subgroup of finite index.

MSC:

20F16 Solvable groups, supersolvable groups
20E36 Automorphisms of infinite groups
20E07 Subgroup theorems; subgroup growth
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