Symplectic Groups

O. T. O’Meara9780821815168, 0-8218-1516-4

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 $mathrm{PSp}_n(mathfrak o)congmathrm{PSp}_{n_1}(mathfrak o_1)Leftrightarrow n=n_1$ and $mathfrak ocongmathfrak o_1$ for dimensions $geq 4$. The new geometric approach used in the book is instrumental in extending the theory from subgroups of $mathrm{PSp})n(ngeq6)$ where it was known to subgroups of $mathrm{P}Gammamathrm{Sp}_n(ngeq4)$ where it is new. There are extensive investigations and several new results on the exceptional behavior of $mathrm{P}Gammamathrm{Sp}_4$ 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.

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Table of contents :
Symplectic Groups……Page 2
Contents……Page 5
Preface……Page 6
Prerequisites and Notation……Page 7
1.1 Alternating Spaces……Page 8
1.2 Projective Transformations……Page 16
1.3 Residues……Page 19
1.4 Transvections……Page 22
1.5 Matrices……Page 24
1.6 Projective Transvections……Page 25
1.7 Some Theorems about SLn……Page 26
1.8 Comments……Page 27
2.1 Generation by Transvections in Spn……Page 29
2.2 Elementary Generation of Spn……Page 37
2.3 Comments……Page 39
3.1 Orders of Symplectic Groups……Page 40
3.2 Centers……Page 42
3.3 Commutator Subgroups……Page 43
3.4 Simplicity Theorems……Page 46
3.5 Comments……Page 48
4.1 Collinear Transformations……Page 49
4.2 Symplectic Collinear Transformations……Page 53
4.3 Hyperbolic Transformations……Page 57
5.1 Groups with Enough Projective Transvections……Page 61
5.2 Preservation of Projective Transvections……Page 62
5.3 The Isomorphism Theorems in General……Page 71
5.4 4-Dimensional Groups in Characteristic 2……Page 74
5.5 Bounding Modules over Integral Domains……Page 97
5.6 The Isomorphism Theorems over Integral Domains……Page 102
5.7 Comments……Page 110
6.1 The Nonisomorphisms……Page 114
Index……Page 124