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Locally graded groups with a nilpotency condition on infinite subsets. (English) Zbl 0982.20019
A group is called locally graded if each non-trivial finitely generated subgroup has a non-trivial finite quotient. Here the authors study locally graded groups in the class \(N(2,k)^*\): this is the class of groups in which every infinite subset contains a pair of elements generating a nilpotent subgroup of class at most \(k\).
The main result of the paper is: Theorem 1. Let \(G\) be a finitely generated, locally graded group in the class \(N(2,k)^*\). Then \(G/Z_c(G)\) is finite for some \(c=c(k)>0\).

MSC:
20F19 Generalizations of solvable and nilpotent groups
20E26 Residual properties and generalizations; residually finite groups
20E25 Local properties of groups
20F05 Generators, relations, and presentations of groups
20F14 Derived series, central series, and generalizations for groups
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