Linear groups and group rings

Abstract

This paper consists of two parts. The first is concerned with free products in linear groups and uses the usual "ping pong" lemma and attractors to prove the results. What is new here is that we allow certain subspaces of V associated with the semisimple and generalized transvection operators to have dimensions larger than 1. The second part is concerned with applications of this machinery to integral groups rings Z[G] of finite groups. We show, for example, that if G is nonabelian of order prime to 6, then Z[G] contains two Bass cyclic units that generate a non-abelian free group.  2005 Elsevier Inc. All rights reserved. Linear operators and attractorsLet F be a field and let | |: F → R + = {r ∈ R: r 0} be an absolute value defined on F . Here R is the field of real numbers and, by definition, we have |ab| = |a| · |b|,

Involutions and free pairs of bicyclic units in integral group rings

“…Thus, in order to handle this one special situation, we require an alternate approach to computing the eigenvalues of m à m. Part (i) below is a generalization of [3,Lemma 4.4] that allows A to be cyclic. As will be apparent, we do not need the full force of this result.…”

Bicyclic units, Bass cyclic units and free groups

“…Using this it is easy to see that the Bass cyclic units of ZG belong to U (ZG). See [7] and [18] Let G be a finite group of order coprime with 6. We have to show that ZG has a free pair formed by a bicyclic unit and a Bass cyclic unit.…”

“…The existence of free pairs in the group of units U (ZG) of the integral group ring ZG, was firstly proved by Hartley and Pickel [8], provided that G is neither abelian nor a Hamiltonian 2-group (equivalently U (ZG) is neither abelian nor finite , which prove that the group generated by the bicyclic and the Bass cyclic units generates a big portion of U (ZG), show that these two types of units have an important role in the structure of U (ZG). As a consequence, several authors have payed attention to the problem of describing the structure of the group generated either by bicyclic units, or Bass cyclic units and more specifically, to the problem of deciding when two bicyclic units or Bass cyclic units form a free pair [3,7,10,16]. The bicyclic units of ZG are the elements of one of the following forms …”

“…Moreover Λ ⊆ L where L is the set of d-th roots of unity. By [7,Lemma 3.5(ii)], if x, y ∈ L then |u k,m (x)| = |u k,m (y)| if and only if x and y are either equal or conjugate. If Z p = X + then Λ has either one element or it has two different conjugate elements.…”

Erratum to “Linear groups and group rings” [J. Algebra 295 (2006) 94–118]

“…The incorrect inequality shows up twice in the paper, once in the proof of [GP,Proposition 1.2] and once in the proof of [GP,Proposition 1.4]. Replace the first paragraph of the proof of [GP,Proposition 1.2] with the following argument that certainly makes more sense.…”