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Characterisation of nilpotent-by-finite groups. (English) Zbl 0959.20034

By a well-known result of B. H. Neumann a group is centre-by-finite if and only if every infinite subset contains a pair of distinct commuting elements. This result has spawned a whole series of articles, the one under review being the latest.
The main result is Theorem 1. Let \(G\) be a finitely generated soluble group. Then \(G\) is nilpotent-by-finite if and only if for every pair of infinite subsets \(X,Y\) there exist \(x\in X\), \(y\in Y\) and positive integers \(m=m(x,y)\), \(n=n(x,y)\) such that \([x,{_ny^m}]=1\).

MSC:

20F18 Nilpotent groups
20F16 Solvable groups, supersolvable groups
20F12 Commutator calculus
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