Serre, Jean-Pierre Linear representations of finite groups. 5th ed., corr. and enlarged with new exercisies. 5ième éd., corr. et augm. de nouveaux exercices. (Représentations linéaires des groupes finis. 5ième éd., corr. et augm. de nouveaux exercices.) (French) Zbl 0926.20003 Collection Méthodes. Paris: Hermann. 182 p. (1998). This is a new edition including new exercises [for a review of the 1st ed. (1967) see Zbl 0189.02603]. The book under review comprises three parts which are different in level as well as in objectives.The first part studies complex representations of finite groups and gives the theory of characters, generalizations to compact groups and several examples in relation with symmetric groups. This part only uses elementary results in linear algebra.The second part also deals with representations in zero characteristic. It uses the framework of group algebras and Grothendieck rings. The main features are induced representations (formula for the character of an induced representation, reciprocity formulas, restriction to subgroups and Mackey’s irreducibility criterion, integrability properties of characters with applications, degrees of irreducible characters, the classical induction theorems of Artin and Brauer completed by Brauer’s splitting field theorem).The third part is an expository of Brauer’s theory of modular representations: the passage from characteristic zero to positive characteristic and vice-versa. The main results there are the fact that the decomposition homomorphism is surjective, that is, the fact that each irreducible representation in positive characteristic is liftable in a convenient Grothendieck group to zero characteristic, and the Fong-Swan theorem which says that each irreducible representation in positive characteristic is liftable to zero characteristic under the assumption that the group is \(p\)-solvable. There is also a section on the Artin and Swan representations, including a proof of their rationality. Reviewer: Anne-Marie Aubert (Paris) Cited in 1 ReviewCited in 15 Documents MSC: 20Cxx Representation theory of groups 20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory 20C15 Ordinary representations and characters 20C20 Modular representations and characters Keywords:complex representations; finite groups; characters; group algebras; Grothendieck rings; induced representations; reciprocity; irreducibility; induction; modular representations Citations:Zbl 0189.02603; Zbl 0205.04001; Zbl 0223.20003; Zbl 0407.20003 PDFBibTeX XMLCite \textit{J.-P. Serre}, Représentations linéaires des groupes finis. 5ième éd., corr. et augm. de nouveaux exercices. Paris: Hermann (1998; Zbl 0926.20003)