# Symplectic Groups

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*O. Timothy O’Meara*

This volume, the sequel to the author's Lectures on Linear
Groups, is the definitive work on the isomorphism theory of symplectic
groups over integral domains. Recently discovered geometric methods which are
both conceptually simple and powerful in their generality are applied to the
symplectic groups for the first time. There is a complete description of the
isomorphisms of the symplectic groups and their congruence subgroups over
integral domains. Illustrative is the theorem \(\mathrm{PSp}_n(\mathfrak
o)\cong\mathrm{PSp}_{n_1}(\mathfrak o_1)\Leftrightarrow n=n_1\) and \(\mathfrak
o\cong\mathfrak o_1\) for dimensions \(\geq 4\). The new geometric
approach used in the book is instrumental in extending the theory from subgroups
of \(\mathrm{PSp})n(n\geq6)\) where it was known to subgroups of
\(\mathrm{P}\Gamma\mathrm{Sp}_n(n\geq4)\) where it is new. There are
extensive investigations and several new results on the exceptional behavior of
\(\mathrm{P}\Gamma\mathrm{Sp}_4\) in characteristic 2.

The author starts essentially from scratch (even the classical simplicity
theorems for \(\mathrm{PSp}_n(F)\) are proved) and the reader need be
familiar with no more than a first course in algebra.

#### Table of Contents

# Table of Contents

## Symplectic Groups

- Contents vii8 free
- Preface ix10 free
- Prerequisites and Notation xi12 free
- Chapter 1. Introduction 114 free
- Chapter 2. Generation Theorems 2336
- Chapter 3. Structure Theorems 3548
- Chapter 4. Symplectic Collinear Transformations 4558
- Chapter 5. The Isomorphisms of Symplectic Groups 5770
- 5.1. Groups with Enough Projective Transvections 5770
- 5.2. Preservation of Projective Transvections 5871
- 5.3. The Isomorphism Theorems in General 6780
- 5.4. 4-Dimensional Groups in Characteristic 2 7083
- 5.5. Bounded Modules over Integral Domains 92105
- 5.6. The Isomorphism Theorems over Integral Domains 98111
- 5.7. Comments 106119

- Chapter 6. The Nonisomorphisms between Linear and Symplectic Groups 111124
- Index 121134