Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T16:16:31.661Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)