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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some results on parafree groups


Author: Yael Roitberg
Journal: Trans. Amer. Math. Soc. 173 (1972), 315-339
MSC: Primary 20E10
MathSciNet review: 0313399
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Abstract: We obtain some theorems concerning parafree groups in certain varieties, which are analogs of corresponding theorems about free groups in these varieties. Our principal results are: (1) A normal subgroup $ N$ of a parafree metabelian group $ P$ of rank $ \geqslant 2$ such that $ N \cdot {\gamma _2}P$ has infinite index in $ P$ is not finitely generated unless it is trivial. (2) If $ x$ and $ y$ are elements of a parafree group $ P$ in any variety containing the variety of all metabelian groups which are independent modulo $ {\gamma _2}P$, then the commutator $ [x,y]$ is not a proper power.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0313399-9
PII: S 0002-9947(1972)0313399-9
Keywords: Parafree groups, varieties of groups, metabelian groups, lower central series, residually nilpotent groups, parabasis, left normed basic commutators
Article copyright: © Copyright 1972 American Mathematical Society