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This is a volume of research articles related to finite groups. Topics covered include the classification of finite simple groups, the theory of p-groups, cohomology of groups, representation theory and the theory of buildings and geometries. As well as more than twenty original papers on the latest developments, which will be of great interest to specialists, the volume contains several expository articles, from which students and non-experts can learn about the present state of knowledge and promising directions for further research. The Finite Groups 2003 conference was held in honor of Joh
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During the last 40 years the theory of finite groups has developed dramatically. The finite simple groups have been classified and are becoming better understood. Tools exist to reduce many questions about arbitrary finite groups to similar questions about simple groups. Since the classification there have been numerous applications of this theory in other branches of mathematics. Finite Group Theory develops the foundations of the theory of finite groups. It can serve as a text for a course on finite groups for students already exposed to a first course in algebra. It could supply the background necessary to begin reading journal articles in the field. For specialists it also provides a reference on the foundations of the subject. This second edition has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises.
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In 1970 Bernd Fischer proved his beautiful theorem classifying the almost simple groups generated by 3-transpositions, and in the process discovered three new sporadic groups, now known as the Fischer groups. Since then, the theory of 3-transposition groups has become an important part of finite simple group theory, but Fischer's work has remained unpublished. 3-Transposition Groups contains the first published proof of Fischer's Theorem, written out completely in one place. Fischer's result, while important and deep (covering a number of complex examples), can be understood by any student with some knowledge of elementary group theory and finite geometry. Thus Part I has minimal prerequisites and could be used as a text for an intermediate level graduate course. Parts II and III are aimed at specialists in finite groups and are a step in the author's program to supply a strong foundation for the theory of sporadic groups.
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Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich and vast area of mathematics. It incorporates tools and methods from algebraic geometry, topology, combinatorics and computer algebra, and has since evolved substantially. With this book, the authors meet the need for a contemporary treatment, complementing in core areas the well-established books of Carter and Digne-Michel. Focusing on applications in finite group theory, the authors gather previously scattered results and allow the reader to get to grips with the large body of literature available on the subject, covering topics such as regular embeddings, the Jordan decomposition of characters, d-Harish-Chandra theory and Lusztig induction for unipotent characters. Requiring only a modest background in algebraic geometry, this useful reference is suitable for beginning graduate students as well as researchers.
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