Algebra 1

This UE develops the classical theory of modules over a principal ring, and the basics 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" of the Licence de Mathématiques of the Université de Montpellier.

1. Bachelor's degree reminders on rings (integral, factorial, principal, Euclidean, body).
2. Modules over a ring: notions of module, algebra over a ring; morphisms, scalar restriction. Factorization and isomorphism theorems. (Sub)Torsion module, free module, finite-type module. Notion of rank. Short exact sequences of modules, module extensions. Product and direct 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 reduction of homomorphisms. Smith normal form, similarity invariants.

4. Representations of finite groups: notion of linear representations of a group, morphism between representations. Representations as modules on the group algebra. Subrepresentation, irreducible representation. Direct sum, tensor product of representations. Complete reducibility. Schur's lemma. Characters, character tables, character orthogonality. Regular representation decomposition. Examples: abelian groups, dihedral groups, symmetric groups.

Hourly volumes :

            CM: 27

            TD : 27

            TP: 0

            Land: 0