We provide a wedge decomposition of the homotopy type of the p-subgroup complex in the case of a finite solvable group G. In particular, this includes a new proof of the result of Quillen which says that this complex is contractible if and only if there is a non-trivial normal p-subgroup in G. We also provide reduction formulas for the G-module structure of the homology groups. Our results are obtained with diagram-methods by gluing the p-subgroup complex of G along the p-subgroup complex of = G/N for a normal p′-subgroup of G.

Let N be a finitely generated normal subgroup of a finitely generated group G. We show that if the trivial subgroup is tame in the factor group G/N, then N is that in G. We also give a short new proof of the fact that quasiconvex subgroups of negatively curved groups are tame. The proof utilizes the concept of the geodesic core of the subgroup and is related to the Dehn algorithm.

We show that diagrammatically reducible two-complexes are characterized by the property: every finity subconmplex of the universal cover collapses to a one-complex. We use this to show that a compact orientable three-manifold with nonempty boundary is Haken if and only if it has a diagrammatically reducible spine. We also formulate an nanlogue of diagrammatic reducibility for higher dimensional complexes. Like Haken three-manifolds, we observe that if n ≥ 4 and M is compact connected n-dimensional manifold with a traingulation, or a spine, satisfying this property, then the interior of the universal cover of M is homeomorphic to Euclidean n-space.

In this paper a necessary and sufficient condition will be given for groups to be ν-isologic, with respect to a given variety of groups ν. Its is also shown that every ν-isologism family of a group contains a ν-Hopfian group. Finally we show that if G is in the variety ν, then every ν-covering group of G is a Hopfian group.

A group G belongs to the class W if G has non-nilpotent proper subgroups and is isomorphic to all of them. The main objects of study are the soluble groups in W that are not finitely generated. It is proved that there are no torsion-free groups of this sort, and a reasonable classification is given in the finite rank case.

The object of this paper is to study the sequence of torsion-free ranks of the quotients by the terms of the lower central series of a finitely generated group. This gives rise to the introduction into the study of finitely generated, residually torison-free nilpotent groups of notions relating to the Gelfand-Kirillov dimension. These notions are explored here. The main result concerning the sequences alluded to is the proof that there are continuously many such sequences.

An example is given of a finitely generated group G for which G ≅ G*cG, where C is infinite cyclic.

This paper reports on a facility of the ANU NQ program for computation of nilpotent groups that satisfy an Engel-n identity. The relevant details of the algorithm are presented together with results on Engel-n groups for moderate values of n.

We study some challenging presentations which arise as groups of deficiency zero. In four cases we settle finiteness: we show that two presentations are for finite groups while two are for infinite groups. Thus we answer three explicit questions in the literature and we provide the first published deficiency zero presentation for a group with derived length seven. The tools we use are coset enumeration and Knuth-Bebdix rewriting, which are well-established as methods for proving finiteness or otherwise of a finitely presented group. We briefly comment on their capabilities and compare their performance.

In previous work [2] calculations of subquadratic second order Dehn functions for various groups were carried out. In this paper we obtain estimates for upper and lower bounds of second order Dehn functions of HNN-extensions, and use these to exhibit an infinite number of different superquadratic second order Dehn functions. At the end of the paper several open questions concerning second order Dehn functions of groups are discussed.

We survey the current state of knowledge of bounds in the restricted Burnside problem. We make two conjectures which are related to the theory of PI-algebras.

Groups, all proper factor-groups of which are hypercentral of finite torsion-free rank, are studied in this article.

In this paper we study groups in which every subgroup is subnormal of defect at most 3. Let G be a group which is either torsion-free or of prime exponent different from 7. We show that every subgroup in G is subnormal of defect at most 3 if and only if G is nilpotent of class at most 3. When G is of exponent 7 the situation is different. While every group of exponent 7, in which every subgroup is subnormal of defect at most 3, is nilpotent of class at most 4, there are examples of such groups with class exactly 4. We also investigate the structure of these groups.

This paper has a twofold purpose. The first is to compute the Euler characteristics of hyperbolic Coxeter groups Ws of level 1 or 2 by a mixture of theoretical and computer aided methods. For groups of level 1 and odd values of |S|, the Euler characteristic is related to the volume of the fundamental region of Ws in hyperbolic space. Secondly we note two methods of imbedding such groups in each other. This reduces the amount of computation needed to determine the Euler characteristics and also reduces the number of essentially different hyperbolic groups that need to be considered.

The wreath product W = A ≀ T, where A ≠ 1, is of type F P2 if and only if T is finite and A is of type F P2.

We outline the classification, up to isometry, of all tetrahedra in hyperbolic space with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncations are all π/2, and those remaining are all submultiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups.

For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary. In particular, for each g ≥ 2, we find a sequence of hyperbolic manifolds with totally geodesic boundary of genus g, which we conjecture to be of least volume among such manifolds.

This paper is concerned with the question of whether n-Engel groups are locally nilpotent. Although this seems unlikely in general, it is shown here that it is the case for the groups in a large class C including all residually soluble and residually finite groups (in fact all groups considered in traditional textbooks on group theory). This follows from the main result that there exist integers c(n), e(n) depending only on n, such that every finitely generated n-Engel group in the class C is both finite-of-exponent-e(n)–by–nilpotent-of-class≤c(n) and nilpotent-of-class≤c(n)–by–finite-of-exponent-e(n). Crucial in the proof is the fact that a finitely generated Engel group has finitely generated commutator subgroup.

We show that the group F discovered by Richard Thompson in 1965 has a subexponential upper bound for its Dehn function. This disproves a conjecture by Gersten. We also prove that F has a regular terminating confluent presentation.

The study of classes of finite groups is divided into two parts. The projective theory studies formations and Schunck classes. The dual injective theory studies Fitting classes. In each type of class a generalisation of Sylow's theorem holds. In this paper we seek further generalisations of Sylow's theorem which hold for classes which are neither injective nor projective, but obey other related properties. Firstly a common framework for the injective and projective theories is constructed. Within the context of this common framework further types of Sylow theorem can then be sought. An example is given of a property which is a simple hybrid of injectivity and projectivity which we will call ‘interjectivity’. A generalised Sylow theorem is then proved in the interjective case.

In this paper a scheme of an ‘economical’ embedding of an arbitrary set of groups without involutions in an infinite group with a proper simple normal subgroup is presented. This scheme is then applied to construction of groups with new properties.