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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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