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Théorie locale des blocs d’un groupe fini. (Local block theory of a finite group). (French) Zbl 0663.20007

Proc. Int. Congr. Math., Berkeley/Calif. 1986, Vol. 1, 360-368 (1987).
[For the entire collection see Zbl 0657.00005.]
One of the roots of local block theory is local group theory which the author describes as the study of the Frobenius category of a finite group G. The objects of the Frobenius category are the p-subgroups of G for a fixed prime p, and its morphisms are the restrictions of inner automorphisms of G. In local block theory the Frobenius category of G is replaced by the Brauer category of a block B of G whose objects are the B-subpairs as defined by J. L. Alperin and the author. Then Brauer’s Third Main Theorem can be interpreted as saying that the Frobenius category of G is isomorphic to the Brauer category of the principal block of G. Several results on the Frobenius category have been generalized to the Brauer category; for example, Sylow’s theorems translate into Brauer’s First Main Theorem. The author explains how, in the known examples, the Brauer category influences the module category of B. These examples suggest some fairly general conjectures, and several of them are mentioned in the text. The paper takes pleasant reading and is strongly recommended for everyone who would like to get an impression of this part of representation theory.
Reviewer: B.Külshammer

MSC:

20C20 Modular representations and characters
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
20-02 Research exposition (monographs, survey articles) pertaining to group theory
16S34 Group rings
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure

Citations:

Zbl 0657.00005