Algebra 1

This course develops the classical theory of modules over a principal ring and the foundations of group representation theory, focusing on finite groups.

Master the basic tools common to all branches of algebra.

A Bachelor's degree in Mathematics.

Recommended prerequisites: the content of the two L3 courses "Groups and Rings 1" and "Groups and Rings 2" from the Bachelor's degree in Mathematics at the University of Montpellier.

1. License reminders on rings (integral, factorial, principal, Euclidean, fields).
2. Modules over a ring: concepts of modules, algebras over a ring; morphisms, scalar restriction. Factorization and isomorphism theorems. Torsion (sub)modules, free modules, finite-type modules. The concept of rank. Exact short sequences of modules, module extensions. Direct product and sum. Tensor product over a field, then over any commutative ring; isomorphism (in finite dimension) between Hom(E,F) and E* ⊗ 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: concept of linear representations of a group, of morphisms between representations. Representations as modules over the algebra of the group. Subrepresentation, irreducible representation. Direct sum, tensor product of representations. Complete reducibility. Schur's lemma. Characters, character tables, orthogonality of characters. The decomposition of the regular representation. Examples: abelian groups, dihedral groups, symmetric groups.

Hourly volumes:

            CM: 27

            TD: 27

            TP: 0

            Land: 0