Our aim in writing this book is to give an up-to-date account of the group-theoretic properties of groups of invertible matrices, where the entries of the matrices lie in a division ring. Our knowledge of this branch of algebra has expanded very rapidly over the last decade. The published part of this material exists only as research papers, and quite a bit, although well-known to the small group of initiates, exists only as private notes. The situation has been reached where, when writing research papers, it is very difficult, or at least very cumbersome, to give the reader adequate references. We hope our book will solve this problem both by providing a textbook for those wishing to study the subject in a systematic way and by providing a convenient reference for research workers. In quite a number of places we have included substantial improvements of both statements and proofs of published theorems.

The importance of linear groups in many branches of mathematics, and even in some parts of physics and chemistry, is wellestablished. By contrast the much more recent theory of skew linear groups has yet to prove itself. Our final chapter indicates some applications to the theory of group algebras, but even there the intervention of skew linear groups is far from decisive.