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Book DetailsGraduate Studies in MathematicsVolume: 39; 2002; 169 ppMSC: Primary 20; 11; Secondary 51
“Classical groups”, named so by Hermann Weyl, are groups of matrices or quotients of matrix groups by small normal subgroups.
Thus the story begins, as Weyl suggested, with “Her Allembracing Majesty”, the general linear group \(GL_n(V)\) of all invertible linear transformations of a vector space \(V\) over a field \(F\). All further groups discussed are either subgroups of \(GL_n(V)\) or closely related quotient groups.
Most of the classical groups consist of invertible linear transformations that respect a bilinear form having some geometric significance, e.g., a quadratic form, a symplectic form, etc. Accordingly, the author develops the required geometric notions, albeit from an algebraic point of view, as the end results should apply to vector spaces over moreorless arbitrary fields, finite or infinite.
The classical groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In recent years, they have played a prominent role in the classification of the finite simple groups.
This text provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles. It is intended for graduate students who have completed standard courses in linear algebra and abstract algebra. The author, L. C. Grove, is a wellknown expert who has published extensively in the subject area.
ReadershipGraduate students and research mathematicians interested in algebraic geometry, group theory, and generalizations.

Table of Contents

Chapters

Chapter 0. Permutation actions

Chapter 1. The basic linear groups

Chapter 2. Bilinear forms

Chapter 3. Symplectic groups

Chapter 4. Symmetric forms and quadratic forms

Chapter 5. Orthogonal geometry (char $F\ne 2$)

Chapter 6. Orthogonal groups (char $F \ne 2$), I

Chapter 7. $O(V)$, $V$ Euclidean

Chapter 8. Clifford algebras (char $F \ne 2$)

Chapter 9. Orthogonal groups (char $F \ne 2$), II

Chapter 10. Hermitian forms and unitary spaces

Chapter 11. Unitary groups

Chapter 12. Orthogonal geometry (char $F = 2$)

Chapter 13. Clifford algebras (char $F = 2$)

Chapter 14. Orthogonal groups (char $F = 2$)

Chapter 15. Further developments


Reviews

Textbook for an indepth course ... provides a nice discussion of various further topics in the study of classical groups and Chevalley groups. ... the text would be great for a class or for students learning the material on their own. The topics are covered in a clean tight fashion with appropriate examples given where possible.
Mathematical Reviews


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“Classical groups”, named so by Hermann Weyl, are groups of matrices or quotients of matrix groups by small normal subgroups.
Thus the story begins, as Weyl suggested, with “Her Allembracing Majesty”, the general linear group \(GL_n(V)\) of all invertible linear transformations of a vector space \(V\) over a field \(F\). All further groups discussed are either subgroups of \(GL_n(V)\) or closely related quotient groups.
Most of the classical groups consist of invertible linear transformations that respect a bilinear form having some geometric significance, e.g., a quadratic form, a symplectic form, etc. Accordingly, the author develops the required geometric notions, albeit from an algebraic point of view, as the end results should apply to vector spaces over moreorless arbitrary fields, finite or infinite.
The classical groups have proved to be important in a wide variety of venues, ranging from physics to geometry and far beyond. In recent years, they have played a prominent role in the classification of the finite simple groups.
This text provides a single source for the basic facts about the classical groups and also includes the required geometrical background information from the first principles. It is intended for graduate students who have completed standard courses in linear algebra and abstract algebra. The author, L. C. Grove, is a wellknown expert who has published extensively in the subject area.
Graduate students and research mathematicians interested in algebraic geometry, group theory, and generalizations.

Chapters

Chapter 0. Permutation actions

Chapter 1. The basic linear groups

Chapter 2. Bilinear forms

Chapter 3. Symplectic groups

Chapter 4. Symmetric forms and quadratic forms

Chapter 5. Orthogonal geometry (char $F\ne 2$)

Chapter 6. Orthogonal groups (char $F \ne 2$), I

Chapter 7. $O(V)$, $V$ Euclidean

Chapter 8. Clifford algebras (char $F \ne 2$)

Chapter 9. Orthogonal groups (char $F \ne 2$), II

Chapter 10. Hermitian forms and unitary spaces

Chapter 11. Unitary groups

Chapter 12. Orthogonal geometry (char $F = 2$)

Chapter 13. Clifford algebras (char $F = 2$)

Chapter 14. Orthogonal groups (char $F = 2$)

Chapter 15. Further developments

Textbook for an indepth course ... provides a nice discussion of various further topics in the study of classical groups and Chevalley groups. ... the text would be great for a class or for students learning the material on their own. The topics are covered in a clean tight fashion with appropriate examples given where possible.
Mathematical Reviews