Blocks of Finite Groups
The Hyperfocal Subalgebra of a Block
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Beschreibung
About 60 years ago, R. Brauer introduced "block theory"; his purpose was to study the group algebra kG of a finite group G over a field k of nonzero characteristic p: any indecomposable two-sided ideal that also is a direct summand of kG determines a G-block.
But the main discovery of Brauer is perhaps the existence of families of infinitely many nonisomorphic groups having a "common block"; i.e., blocks having mutually isomorphic "source algebras".
In this book, based on a course given by the author at Wuhan University in 1999, all the concepts mentioned are introduced, and all the proofs are developed completely. Its main purpose is the proof of the existence and the uniqueness of the "hyperfocal subalgebra" in the source algebra. This result seems fundamental in block theory; for instance, the structure of the source algebra of a nilpotent block, an important fact in block theory, can be obtained as a corollary.
The exceptional layout of this bilingual edition featuring 2 columns per page (one English, one Chinese) sharing the displayed mathematical formulas is the joint achievement of the author and A. Arabia.
The exceptional layout of this bilingual edition featurinmg two colums per page (one English, one Chinese) sharing the displayed mathematical formula, is the joint achievement of the author and A. Arabia. Includes supplementary material: sn.pub/extras
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Group algebra block hyperfocal algebra source algebra
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From the reviews:
"The author’s purpose here is to provide a better understanding of the subject, and to make the proofs understandable by students in group theory. The book, which is written in both Chinese (in simple characters) and English, consists of 16 sections. All the concepts are carefully introduced, and the proofs are complete and given in detail, assuming only knowledge of Wedderburn’s theorems, Nakayama’s Lemma and other basic algebraic topics." (Jian Bei An, Mathematical Reviews, 2003 j)
"The book … contains an exposition of the author’s main result on the hyperfocal subalgebra of a block. … The book is bilingual; each page consists of two columns, one with the Chinese and one with the English text. The author has made an effort to make his exposition self-contained … . Also, the original proof of the main result has been modified and improved at several points. One of the new concepts that is studied … is that of a divisor on a G-algebra." (Burkhard Külshammer, Zentralblatt MATH, Vol. 1002 (2), 2003)
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