The aim of the course is to introduce students to the theory of the characters of ordinary representations of finite groups and its application to study the structure of finite groups. It is planned to discuss in addition to the classical aspects of character theory (orthogonality relations, character induction, Frobenius reciprocity law, the number of irreducible complex characters of a finite group, etc.) also Burnside's theorem on the solvability of groups of order $p^{\alpha}q^{\beta}$, Frobenius' theorem on Frobenius groups, Brauer's characterization of the ring of generalized complex characters and, as a consequence, Brauer's theorem on the realizability of irreducible complex representations of a finite group of order $n$ in the $n$-circular extension of the field of rational numbers. An infroduction into the theory of the Schur index of complex characters will be given. An estimate of the decomposition fields of finite groups based on the Goldschmidt-Isaacs theorem will be given.

Financial support. The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2022-265).

Program

Lecturer
Kiselev Denis Dmitrievich

Institutions
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center