E. F. Robertson

This two-volume book contains selected papers from the international conference ‘Groups 1993 Galway/St Andrews’ which was held at University College, Galway in August 1993. The wealth and diversity of group theory is represented in these two volumes. Five main lecture courses were given at the conference. These were ‘Geometry, Steinberg representations and complexity’ by J. L. Alperin (Chicago), ‘Rickard equivalences and block theory’ by M. Broué (ENS, Paris), ‘Cohomological finiteness conditions’, by P. H. Kropholler (Queen Mary and Westfield College, London), ‘Counting finite index subgroups’, by A. Lubotzky (Hebrew University, Jerusalem), ‘Lie methods in group theory’ by E. I. Zel‘manov (University of Wisconsin at Madison). Articles based on their lectures, in one case co-authored, form a substantial part of the Proceedings. Another main feature of the conference was a GAP workshop jointly run by J. Neubüser and M. Schönert (Rheinisch-Westfalische Technische Hochschole, Aachen). Two articles by Professor Neubüser, one co-authored, appear in the Proceedings. The other articles in the two volumes comprise both refereed survey and research articles contributed by other conference participants. As with the Proceedings of the earlier ‘Groups-St Andrews’ conferences it is hoped that the articles in these Proceedings will, with their many references, prove valuable both to experienced researchers and also to new postgraduates interested in group theory.

This two-volume book contains selected papers from the international conference 'Groups 1993 Galway/St Andrews' which was held at University College Galway in August 1993. The wealth and diversity of group theory is represented in these two volumes. Five main lecture courses were given at the conference. These were 'Geometry, Steinberg representations and complexity' by J. L. Alperin (Chicago), 'Rickard equivalences and block theory' by M. Broué (ENS, Paris), 'Cohomological finiteness conditions' by P. H. Kropholler (QMW, London), 'Counting finite index subgroups' by A. Lubotzky (Hebrew University, Jerusalem), 'Lie methods in group theory' by E. I. Zel'manov (University of Wisconsin at Madison). Articles based on their lectures, in one case co-authored, form a substantial part of the Proceedings. Another main feature of the conference was a GAP workshop jointly run by J. Neubüser and M. Schönert (RWTH, Aachen). Two articles by Professor Neubüser, one co-authored, appear in the Proceedings. The other articles in the two volumes comprise both refereed survey and research articles contributed by other conference participants. As with the Proceedings of the earlier 'Groups-St Andrews' conferences it is hoped that the articles in these Proceedings will, with their many references, prove valuable both to experienced researchers and also to new postgraduates interested in group theory.

Basic Linear Algebra is a text for first year students leading from concrete examples to abstract theorems, via tutorial-type exercises. More exercises (of the kind a student may expect in examination papers) are grouped at the end of each section. The book covers the most important basics of any first course on linear algebra, explaining the algebra of matrices with applications to analytic geometry, systems of linear equations, difference equations and complex numbers. Linear equations are treated via Hermite normal forms which provides a successful and concrete explanation of the notion of linear independence. Another important highlight is the connection between linear mappings and matrices leading to the change of basis theorem which opens the door to the notion of similarity. This new and revised edition features additional exercises and coverage of Cramer's rule (omitted from the first edition). However, it is the new, extra chapter on computer assistance that will be ofparticular interest to readers: this will take the form of a tutorial on the use of the "LinearAlgebra" package in MAPLE 7 and will deal with all the aspects of linear algebra developed within the book.

Most of the introductory courses on linear algebra develop the basic theory of finite­ dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix. Generally speaking, such courses culminate in the diagonalisation of certain matrices and the application of this process to various situations. Such is the case, for example, in our previous SUMS volume Basic Linear Algebra. The present text is a continuation of that volume, and has the objective of introducing the reader to more advanced properties of vector spaces and linear mappings, and consequently of matrices. For readers who are not familiar with the contents of Basic Linear Algebra we provide an introductory chapter that consists of a compact summary of the prerequisites for the present volume. In order to consolidate the student's understanding we have included a large num­ ber of illustrative and worked examples, as well as many exercises that are strategi­ cally placed throughout the text. Solutions to the exercises are also provided. Many applications of linear algebra require careful, and at times rather tedious, calculations by hand. Very often these are subject to error, so the assistance of a com­ puter is welcome. As far as computation in algebra is concerned, there are several packages available. Here we include, in the spirit of a tutorial, a chapter that gives 1 a brief introduction to the use of MAPLE in dealing with numerical and algebraic problems in linear algebra.