J. R. J. Groves

A conjecture of Lennox and Wiegold concerning supersoluble groups

We prove a conjecture of Lennox and Wiegold that a finitely generated soluble group, in which every infinite subset contains two elements generating a supersoluble group, is finite-by-supersoluble.

Tensor powers of modules over finitely generated abelian groups

In an earlier paper [ 11, the authors investigated, as a major component of their final result, a special case of the following question. Let R be a commutative ring with unity, G a finitely generated abelian group and A4 a finitely generated RG-module. Consider the tensor power @“, M as an RG-module via the diagonal action of G. Under what conditions is @“, M a finitely generated RG-module? The answer obtained in [ 1 ] was for the case when R is a field and used an invariant Z;M introduced by the first author and Strebel in [3]. We briefly describe CL. Firstly, if n is the torsion-free rank of G, we can identify Hom(G, [w) with [w”. Defining two elements of Hom(G, Iw) to be equivalent if they differ by a positive real multiple, we obtain a natural projection of Hom(G, [w)\(O) onto the sphere s” I. Denote the equivalence class of XE Hom(G, 1w) by [x] and define GcXI = G,= { ge G: x(g) 30). Then Z:, is the subset of S”’ consisting of all those [x] for which M is a finitely generated RG,-module and CL is the complement of C, in S”‘. We say that M is k-tame if there do not exist [x, I,..., [xk ] in Zh such that x1 + “. + xk = 0. Then the partial answer referred to above (Theorem 3.4 of [l]) is

Minimal length normal forms for some soluble groups

Abstract It is shown that it is not possible to obtain a regular set of length-minimal normal forms for a class of infinite soluble groups equipped with natural generating sets. The class of groups includes the soluble Baumslag-Solitar groups with their natural generating sets.

Finite presentation of abelian-by-finite-dimensional Lie algebras

In this paper we shall prove the main part ((3) implies (1)) of the following theorem. The proof of the theorem will be completed in a forthcoming paper [ 6 ].

Some finitely presented nilpotent-by-abelian groups

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

Abelian-by-polycyclic groups of homological type FP3

Abstract We show that abelian-by-polycyclic groups of homological type FP3 are virtually nilpotent-by-abelian; this generalizes earlier results of C.J.B. Brookes. We apply this to show that if G is the split extension of A by Q with A abelian, Q polycyclic and G of homological type FP3, then every quotient of G also has homological type FP3.

Remarks on a technique of Dunwoody

Abstract It is shown that Dunwoodys proof of the accessibility of almost finitely presented groups extends to a larger class of groups. This class is defined by a condition on the homology of a space on which the group acts. An equivalent algebraic condition is also given.

Finitely presented centre-by-metabelian Lie algebras

It is shown that finitely presented centre-by-metabelian Lie algebras are Abelian-by-finite-dimensional.

Fitting quotients of finitely presented abelian-by-nilpotent groups

We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

Nilpotent-by-abelian Lie algebras of type FP

Let L be a finitely generated Lie algebra which is a split extension of a free nilpotent Lie algebra N by a finite dimensional abelian Lie algebra. Let V denote the quotient of N by its commutator subalgebra; we can regard V as a module for L / N . We discuss the relationship between the homological finiteness properties of V and those of L . In particular, we show that if L has type FP m and N has class c then V is 1 + c ( m − 1)-tame (equivalently, the (1 + c ( m − 1))th tensor power of V is finitely generated under the diagonal action of L / N ).