# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] C. J. B.Brookes, The primitivity of group rings of soluble groups with trivial periodic radical. To appear in J. London Math. Soc. · Zbl 0532.16008 [2] C. J. B.Brookes and K. A.Brown, Primitive group rings and Noetherian rings of quotients. Preprint. · Zbl 0562.16005 [3] K. A. Brown, Primitive group rings of soluble groups. Arch. Math.36, 404-413 (1981). · Zbl 0464.16009 · doi:10.1007/BF01223718 [4] P.Hall, The Edmonton notes on nilpotent groups. London 1969. · Zbl 0211.34201 [5] A. I. Mal’cev, On certain classes of infinite soluble groups. Mat. Sb.28, 567-588 (1951) ? Amer. Math. Soc. Transl. (2)2, 1-21 (1956). [6] S. Moran, A subgroup theorem for free nilpotent groups. Trans. Amer. Math. Soc.103, 495-515 (1962). · Zbl 0199.05901 · doi:10.1090/S0002-9947-1962-0155886-6 [7] D. J. S.Robinson, Finiteness conditions and generalised soluble groups, Vol. II. Berlin 1972. · Zbl 0243.20032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.