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Algebraic groups are treated in this volume from a group theoretical point of view and the obtained results are compared with the analogous issues in the theory of Lie groups. The main body of the text is devoted to a classification of algebraic groups and Lie groups having only few subgroups or few factor groups of different type. In particular, the diversity of the nature of algebraic groups over fields of positive characteristic and over fields of characteristic zero is emphasized. This is revealed by the plethora of three-dimensional unipotent algebraic groups over a perfect field of positive characteristic, as well as, by many concrete examples which cover an area systematically. In the final section, algebraic groups and Lie groups having many closed normal subgroups are determined.
List of contents
Prerequisites.- Extensions.- Groups of Extreme Nilpotency Class.- Chains.- Groups with Few Types of Isogenous Factors.- Three-Dimensional Affine Groups.- Normality of Subgroups.
About the author
Prof. Dr. Karl Strambach ist seit 1972 ordentlicher Professor an der Universität Erlangen. Nach seiner Habilitation arbeitete er zunächst als Universitätsdozent und Wissenschaftlicher Rat an der Universität Tübingen sowie als Professor an der Universität Kiel. 2007 wurde ihm vonder Universität Debrecen die Ehrendoktorwürde verliehen.
Summary
Algebraic groups are treated in this volume from a group theoretical point of view and the obtained results are compared with the analogous issues in the theory of Lie groups. The main body of the text is devoted to a classification of algebraic groups and Lie groups having only few subgroups or few factor groups of different type. In particular, the diversity of the nature of algebraic groups over fields of positive characteristic and over fields of characteristic zero is emphasized. This is revealed by the plethora of three-dimensional unipotent algebraic groups over a perfect field of positive characteristic, as well as, by many concrete examples which cover an area systematically. In the final section, algebraic groups and Lie groups having many closed normal subgroups are determined.
Additional text
From the reviews:"This is a self-contained carefully written lecture note on algebraic groups as well as real and complex Lie groups. For algebraic groups the authors stress the difference between groups over fields of characteristic zero and characteristic p. … Overall the notes provide an excellent introduction to the subject and this research monograph presents remarkable and powerful results. It is a good reference for researchers working on groups outside the family of reductive groups." (Jorge A. Vargas, Mathematical Reviews, Issue 2009 g)“The authors study both algebraic groups and real and complex Lie groups by imposing on their connected subgroups the analogues of familiar conditions in abstract group theory. … This book serves as a valuable resource to those working in the field of algebraic groups and as a good place to learn about this aspect of that field.” (Ernest L. Stitzinger, Zentralblatt MATH, Vol. 1176, 2010)
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From the reviews:
"This is a self-contained carefully written lecture note on algebraic groups as well as real and complex Lie groups. For algebraic groups the authors stress the difference between groups over fields of characteristic zero and characteristic p. ... Overall the notes provide an excellent introduction to the subject and this research monograph presents remarkable and powerful results. It is a good reference for researchers working on groups outside the family of reductive groups." (Jorge A. Vargas, Mathematical Reviews, Issue 2009 g)
"The authors study both algebraic groups and real and complex Lie groups by imposing on their connected subgroups the analogues of familiar conditions in abstract group theory. ... This book serves as a valuable resource to those working in the field of algebraic groups and as a good place to learn about this aspect of that field." (Ernest L. Stitzinger, Zentralblatt MATH, Vol. 1176, 2010)
