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
CONTENT
1. Introduction -- 2. Lifting Idempotents -- 3. Points of the O-algebras and Multiplicity of the Points -- 4. Divisors on N-interior G-algebras -- 5. Restriction and Induction of Divisors -- 6. Local Pointed Groups on N-interior G-algebras -- 7. On Greenโ{128}{153}s Indecomposability Theorem -- 8. Fusions in N-interior G-algebras -- 9. N-interior G-algebras through G-interior Algebras -- 10. Pointed Groups on the Group Algebra -- 11. Fusion ?-algebra of a Block -- 12. Source Algebras of Blocks -- 13. Local Structure of the Hyperfocal Subalgebra -- 14. Uniqueness of the Hyperfocal Subalgebra -- 15. Existence of the Hyperfocal Subalgebra -- 16. On the Exponential and Logarithmic Functions in O-algebras -- References
