Brauer had already introduced the defect of a block and opened the way towards a classification by solving all the problems in defects zero and one, and by providing some evidence for the finiteness of the set of blocks with a given defect. In 1959 he discovered the defect group, and in 1964 Dade determined the blocks with cyclic defect groups.
In 1978 Alperin and BrouA(c) discovered the Brauer category, and BrouA(c) and the author determined the blocks having a nilpotent Brauer category. In 1979, the author discovered the source algebra which determines all the other current invariants, representing faithfully the block a " and found its structure in the nilpotent blocks. Recently, the discovery by Rickard that all blocks with the same cyclic defect group and the same Brauer category have the same homotopic category focussed great interest on the new, loose relationship between blocks called Rickard equivalence.
This book describes the source algebra of a block from the source algebra of a Rickard equivalent block and the source of the Rickard equivalence.

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.

I1 More than one hundred years ago, Georg Frobenius [26] proved his remarkable theorem a?rming that, for a primep and a ?nite groupG, if the quotient of the normalizer by the centralizer of anyp-subgroup ofG is a p-group then, up to a normal subgroup of order prime top,G is ap-group. Ofcourse,itwouldbeananachronismtopretendthatFrobenius,when doing this theorem, was thinking the category - notedF in the sequel - G where the objects are thep-subgroups ofG and the morphisms are the group homomorphisms between them which are induced by theG-conjugation. Yet Frobenius' hypothesis is truly meaningful in this category. I2 Fifty years ago, John Thompson [57] built his seminal proof of the nilpotencyoftheso-called Frobeniuskernelofa FrobeniusgroupGwithar- ments - at that time completely new - which might be rewritten in terms ofF; indeed, some time later, following these kind of arguments, George G Glauberman [27] proved that, under some - rather strong - hypothesis onG, the normalizerNofasuitablenontrivial p-subgroup ofG controls fusion inG, which amounts to saying that the inclusionN?G induces an ? equivalence of categoriesF =F .

Brauer had already introduced the defect of a block and opened the way towards a classification by solving all the problems in defects zero and one, and by providing some evidence for the finiteness of the set of blocks with a given defect. In 1959 he discovered the defect group, and in 1964 Dade determined the blocks with cyclic defect groups.
In 1978 Alperin and Broue discovered the Brauer category, and Broue and the author determined the blocks having a nilpotent Brauer category. In 1979, the author discovered the source algebra which determines all the other current invariants, representing faithfully the block and found its structure in the nilpotent blocks. Recently, the discovery by Rickard that all blocks with the same cyclic defect group and the same Brauer category have the same homotopic category focussed great interest on the new, loose relationship between blocks called Rickard equivalence.
This book describes the source algebra of a block from the source algebra of a Rickard equivalent block and the source of the Rickard equivalence."

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.