Let KG be the group ring of the group G over the field K and U(KG) its unit group. When G is finite we derive conditions which imply that every noncentral subnormal subgroup of U(KG) contains a free group of rank two. We also show that residual nilpotence of U(KG) coincides with nilpotence, this being no longer true if G is infinite.We can answer partially the following question: when is G sub-normal in U(KG)?

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|,

No abstract

Let A be a quaternion algebra over a commutative unital ring. We find sufficient conditions for pairs of units of A to generate a free group. Using the Ž . well-known isomorphism between SO 3, ‫ޒ‬ and the group of real quaternions of norm 1, we obtain free groups of rotations of the Euclidean 3-space. Specialization techniques allow us to find similar free subgroups in skew polynomial rings. A Ž consequence is the following: let kG be the group algebra of a residually torsion-. free nilpotent group G over a field k whose characteristic is not 2. If x and y are any pair of noncommuting elements of G, and c, d g k U then 1 q cx and 1 q dy generate a free subgroup of the Malcev᎐Neumann field of fractions of kG. ᮊ 1999 Academic Press U