# Lectures on Linear Groups

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*O. T. O’Meara*

A co-publication of the AMS and CBMS

The notes in this volume evolved from lectures at the California
Institute of Technology during the spring of 1968, from ten survey lectures on
classical and Chevalley groups at an NSF Regional Conference at Arizona State
University in March 1973, and from lectures on linear groups at the University
of Notre Dame in the fall of 1973.

The author's goal in these expository lectures was to explain the
isomorphism theory of linear groups over integral domains as illustrated by the
theorem
\[\mathrm{PSL}_n({\mathfrak o})\cong\mathrm{PSL}_{n_1}({\mathfrak o}_1)\Longleftrightarrow
n=n_1\quad\mathrm{and}\quad{\mathfrak o}\cong{\mathfrak o}_1\]
for dimensions \(\geq 3\). The theory that follows is typical of much of
the research of the last decade on the isomorphisms of the classical groups
over rings. The author starts from scratch, assuming only basic facts from a
first course in algebra. The classical theorem on the simplicity of
\(\mathrm{PSL}_n(F)\) is proved, and whatever is needed from projective
geometry is developed. Since the primary interest is in integral domains, the
treatment is commutative throughout. In reorganizing the literature for these
lectures the author extends the known theory from groups of linear
transformations to groups of collinear transformations, and also improves the
isomorphism theory from dimensions \(\geq 3\).

#### Reviews & Endorsements

Fine expository work, in a concisely-written volume, of questions relevant to ‘general’ linear groups—that is, essentially the group \(\mathrm{GL}_n({\mathfrak o})\) of invertible matrices of the order \(n\) over the ring \({\mathfrak o}\) of integers, its distinguished subgroups, and its factor groups.

-- J. Dieudonné, Mathematical Reviews

#### Table of Contents

# Table of Contents

## Lectures on Linear Groups

- Cover v6 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Preface vii8 free
- Prerequisites and Notation 110 free
- Chapter 1: Introduction 211
- Chapter 2: Generation Theorems 1423
- Chapter 3: Structure Theory 2029
- Chapter 4: Collinear Transformations and Projective Geometry 2837
- Chapter 5: The Isomorphisms of Linear Groups 4049
- 5.1. Preliminaries 4049
- 5.2. Full groups 4251
- 5.3. CDC in the Linear Case 4554
- 5.4. Preservation of Projective Transvections in the Linear Case 5160
- 5.5. The Isomorphism Theorems in General 5665
- 5.6. The Isomorphism Theorems over Fields 6574
- 5.7. The Isomorphism Theorems over Integral Domains 7685
- 5.8. Comments 8392

- Index 8695
- Back Cover Back Cover197