Introduction

Abstract

The purpose of this book is to give an account of the fundamental algebraic properties of the classical groups over rings. This theory was initiated by C. Jordan in the 1860’s with his investigations of certain matrix groups with coefficients in a finite prime field. Towards the turn of the century L. E. Dickson expanded on this work with his systematic study of matrix groups over arbitrary finite fields. The last landmark in the theory was Dieudonné’s “La Géométrie des Groupes Classiques” which appeared in the mid 1950’s. In this important volume Dieudonné considered the following families of groups over arbitrary fields and division rings: general linear groups, symplectic groups, unitary groups and orthogonal groups. He then pursued the following themes for these “classical” groups. He analyzed their generators, he studied their subgroups and quotient groups, he established the simplicity of their projective commutator subgroups, and he described their isomorphisms.