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Lectures on Linear Groups
A co-publication of the AMS and CBMS.
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CBMS Regional Conference Series in Mathematics
1974; 87 pp
Number: 22
Reprint/Revision History:
reprinted with corrections 1977; third printing 1988
ISBN-10: 0-8218-9840-X
ISBN-13: 978-0-8218-9840-6
List Price: US$27 Member Price: US$21.60
All Individuals: US\$21.60
Order Code: CBMS/22.D

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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

"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