Infinite Groups and Group Rings

The following sections are included:

• PREFACE

• CONTENTS

Let M be a maximal subgroup of the group G. The starting point for our discussion is a result due to Herstein [3], which states that, for finite (G, if M is abelian then G is soluble. This result is quite deep, depending as it does on the theory of Frobenius groups (and hence on character theory). The basic idea of the proof is straightforward: factoring (as one may) by the core of M in G, one arrives at the situation where M ∩ M^x = 1 for all x ∉ M. Thus M is a Frobenius complement and has a Frobenius kernel N, say, which is nilpotent by Thompson's theorem…

Let G be a finite p-group, p a prime, and let kG denote the group algebra of G over the field k with p elements. Given the multiplication table for such a group (generated by Cayley) we have written programs in C to manipulate elements of kG and to generate bases for certain ideals and subspaces. In particular, the Lie subspaces and ideals can be computed. It is possible to investigate the isomorphism problem in an interactive way. The work has been done on a Sun SparcStation.

In 1962 a paper by G. Baumslag was published proving that two–generator subgroups of the free product of two free groups amalgamated with certain subgroups are free of rank 2. Over the last thirty years this result has been extended by various authors in different directions, and this paper is to be seen in this context. It provides an “HNN–analogue” of the amalgamated product situation and some applications.

This paper gives group theoretical characterizations of the universal theory of the non-Abelian free groups. In particular for a non-Abelian residually free group G, necessary and sufficient conditions are established for G to be a model of the universal theory of the non-Abelian free groups.

In this paper, we let G be a finite group and R an unramified extension of the integers localized at a prime. We describe the pullback structure of a block of RG with cyclic, normal defect group. This pullback structure generalizes the special quasi-triad found in [HK1] and [HK2].

Let ZG be the integral group ring of the infinite dihedral group. It is shown that any torsion unit of ZG, having augmentation one, is conjugate in QG to an element of G. This result is applied to deduce a result of Wallace that any normalized automorphism of ZG is composed of an automorphism of G and a conjugation by a suitable element of QG.

We prove a general theorem about the endomorphism structure of certain affine triangles of groups. We then apply this result to some particular examples, for instance PSL₂(Z[ω]) (where ω satisfies ω² = ω – 1) and a set of groups defined by A. Haefliger.

Let K be a field of characteristic p > 0 and let G be a locally finite group with p-subgroups of bounded order. We prove that JK[G], the Jacobson radical of the group algebra K[G], is nilpotent. In particular, K[G] is semiprimitive if and only if G has no finite normal subgroup of order divisible by p. The argument here is a mixture of group ring and group theoretic techniques and requires that we deal more generally with twisted group algebras. Furthermore, the proof ultimately depends upon consequences of the classification of the finite simple groups.

A finitely generated group which is virtually a direct product of polycyclic and free groups has a full automorphism group which is finitely presented and virtually of finite cohomological dimension.

Let G be an infinite group and m a fixed positive integer. If for every n = 2^m infinite subsets Y₁, Y₂,…, Y_n of G there exist elements y_i in Y_i, i = 1, 2,…, n such that the subgroup generated by {y₁, J₂,…, y_n} is soluble of length at most m, then the group G is soluble of length at most m. We shall indicate a proof of this result which uses group theoretical methods together with combinatorial tools such as Ramsey's Theorem.

A survey is given of recent results on the structure of a group G which has a factorization of the form G = AB = BC = CA where A, B, C are abelian subgroups.

The following sections are included:

• Introduction

• Preliminary remarks

• Vertical Heegaard decompositions

• Proof of Theorem (1.1)

• References

The following sections are included:

• Introduction

• References

The following sections are included:

• Introduction

• Maximal subgroups of infinite symmetric groups

• The number of orbits on

• Groups which act transitively on moieties

• REFERENCES

The following sections are included:

• LIST OF PARTICIPANTS AT THE SPECIAL SESSION