The theory of the representations and characters of finite groups is central both to algebra and group theory. The tools that the theory develops enable us to reach a deeper understanding of the structure of a finite group. We shall study classical ordinary, or complex, representations and characters, and then develop briefly the main theorems and ideas of modular representations and characters. Modular representation theory deals with representations of finite groups over a field of prime characteristic, and their connections and relations with the ordinary representations. The resulting beautiful theory which was developed by Brauer, gives additional insights into the structure of the group.
Group algebras, Maschke's theorem, structure of finite-dimensional semi-simple algebras over a field, centre of the group algebra. Ordinary (complex) irreducible representations and characters of finite groups, character tables, orthogonality relations. Central characters and algebraic integers. Burnside's theorem. Induced characters, Frobenius reciprocity. Modular representations of finite groups.Brauer characters, blocks and decomposition numbers. Defect groups, the Brauer correspondence, Brauer's first main theorem. Blocks of cyclic defect and Brauer trees, Dade's theorem.