Algebra 1

This unit expands on the classical theory of modules over a principal ring and the fundamentals of group representation theory, focusing on finite groups.

Master the basic tools common to all branches of algebra.

A bachelor's degree program in mathematics.

Recommended prerequisites: the material covered in the two third-year courses, “Groups and Rings 1” and “Groups and Rings 2,” in the Bachelor of Science in Mathematics program at the University of Montpellier.

1. Review of Ring Theory (integral rings, factorial rings, principal rings, Euclidean rings, fields).
2. Modules over a ring: concepts of modules and algebras over a ring; morphisms; scalar restriction. Factorization and isomorphism theorems. Torsion (sub)modules, free modules, finite-type modules. The concept of rank. Short exact sequences of modules, module extensions. Direct product and direct sum. Tensor product over a field, then over any commutative ring; isomorphism (in finite dimensions) between $Hom(E,F)$ and $E* \times F$.
3. Structure of finite-type modules over a principal ring. Application to abelian groups; application to the reduction of homomorphisms. Smith normal form; similarity invariants.

4. Representations of finite groups: the concept of linear representations of a group and morphisms between representations. Representations as modules over the group algebra. Subrepresentations, irreducible representations. Direct sum, tensor product of representations. Complete reducibility. Schur’s lemma. Characters, character tables, orthogonality of characters. The decomposition of a regular representation. Examples: Abelian groups, dihedral groups, symmetric groups.

Hourly volumes:

            CM: 27

            TD: 27

            TP: 0

            Land: 0