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PRESENTATIONS OF FREE ABELIAN-BY-(NILPOTENT OF CLASS 2) GROUPS

Published online by Cambridge University Press:  01 March 1998

MARTIN J. EVANS
MARTIN J. EVANS
Affiliation:
Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA
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Abstract

Let Fn, Mn and Un denote the free group of rank n, the free metabelian group of rank n, and the free abelian-by-(nilpotent of class 2) group of rank n, respectively. Thus Mn≅ Fn/F″″n and Un≅Fn/ [γ3(Fn), γ3(Fn)], where γ3(Fn) =[F′n, Fn], the third term of the lower central series of Fn.

Consider an arbitrary epimorphism θ[ratio ]Fn[Rarr ]Fk, where k<n. A routine argument (see [10, Theorem 3.3]) using the Nielsen reduction process shows that there exists a free basis {y1, y2, …, yn} of Fn such that kerθ is the normal closure in Fn of {y1, y2, …, yn−k}. A similar result has been obtained for free metabelian groups by C. K. Gupta, N. D. Gupta and G. A. Noskov [5]. Indeed, it follows immediately from [5, Theorem 3.1] that if k<n and θ[ratio ]Mn[Rarr ]Mk is an epimorphism, then there exists a free basis {m1, m2, …, mn} of Mn such that kerθ is the normal closure in Mn of {m1, m2, …, mn−k}.

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 30 , Issue 2 , March 1998 , pp. 136 - 144
Copyright
© The London Mathematical Society 1998

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