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Locally nilpotent groups with an intersection property. (English) Zbl 0544.20035
Let B be a subgroup of \(\zeta_ r(G)\), the rth term of the upper central series of a torsion-free group G. Suppose all abelian subgroups of G having trivial intersection with B are of finite rank. Then all subgroups of G have finite upper central height. Moreover if G is locally nilpotent then it is nilpotent and has a normal subgroup \(G_ 1\) of nilpotency class at most 2r, with \(G/G_ 1\) torsion-free of finite rank. Any free nilpotent group G of class 2r has the property that all abelian subgroups intersecting trivially with \(\zeta_ r(G)\) are of finite rank.

MSC:
20F19 Generalizations of solvable and nilpotent groups
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
20F14 Derived series, central series, and generalizations for groups
20F18 Nilpotent groups
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