Algebra 1

This UE develops the classical theory of modules on a principal ring, and the basics of the theory of representations of groups by concentrating 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 University of Montpellier.

1. Reminder of the Bachelor's degree on rings (integral, factorial, principal, Euclidean, body).
2. Modules on a ring: notions of module, algebra on a ring; morphisms, restriction of scalars. Factorization and isomorphism theorems. (Sub)Torsion module, free module, module of finite type. Notion of rank. Short exact sequence of modules, extensions of modules. 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 modules of finite type over a principal ring. Application to abelian groups, application to the 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, orthogonality of characters. Decomposition of the regular representation. Examples: abelian groups, dihedral groups, symmetric groups.

Hourly volumes:

            CM : 27

            TD : 27

            TP : 0

            Land : 0