Instructor: Dr. József PELIKÁN

Text: I. Martin Isaacs, Character Theory of Finite Groups

Prerequisite: An introductory algebra course covering the basic definitions and theorems on groups, rings and modules.

Course description: This is a course on the representation theory of finite groups, with emphasis on the use and application of group characters.

Topics:
Preliminaries from group theory: normalizer, centralizer, conjugacy classes. Center, commutator subgroup. Sylow's theorems.
Group representations, the group algebra. Schur's lemma. Maschke's theorem. Wedderburn's theorem on semisimple algebras. Irreducible and completely reducible representations.
Characters. The orthogonality relations. Determination of center, commutator subgroup, normal subgroups from the character table. Algebraic integers. Burnside's p^a q^b theorem. Theorems on character degrees. Products of characters.
Induced representations and characters. M-groups, Taketa's theorem. Frobenius groups. Frobenius' theorem on Frobenius kernels. TI-sets and exceptional characters. Characterizations of groups by 2-Sylow subgroups and centralizers of involutions.
Normal subgroups. Clifford's Theorem