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- W2083828758 abstract "This paper gives some sufficient conditions for the finite presentability of nilpotent-by-abelian groups. The general question of deciding whether a particular soluble group admits a finite presentation has received considerable attention in recent years; see the review article of Strebel [lo] for a discussion of the work to 1984. In particular, we want to draw attention to three papers particularly relevant to the topic of this paper. Firstly, Bieri and Strebel have given in [4] a necessary and sufficient condition for the finite presentation of a metabelian group; the present work is based on their methods. The sufficient condition was extended by McIsaac in [7] to a condition for the finite presentation of a split extension of a group of nilpotency class 2 by an abelian group. In a major work, Abels [2] deals with finite presentability of arithmetic groups. His work shows, for example, that soluble arithmetic groups corresponding to connected algebraic groups are finitely presented if and only if the metabelian quotient is finitely presented and the multiplier (the second homology with trivial coefficients) of the derived group is finitely generated as a module for the group. This therefore gives a superb answer to the question for arithmetic groups but it is not clear to what extent such results are available for general abstract groups. We return briefly to this later in this introduction. The aim of this paper is to carry the sufficient condition given by McIsaac considerably further. We remove the restriction that the extension involved be split and widen the condition considerably. In particular we make use of the Jacobi identity which did not figure in McIsaac’s treatment. We also extend the results to the case where the nilpotent normal subgroup has arbitrary nilpotency class. To describe our results, we need to introduce some notation and to" @default.
- W2083828758 created "2016-06-24" @default.
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- W2083828758 date "1991-12-01" @default.
- W2083828758 modified "2023-09-29" @default.
- W2083828758 title "Some finitely presented nilpotent-by-abelian groups" @default.
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- W2083828758 doi "https://doi.org/10.1016/0021-8693(91)90133-s" @default.
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