TD Algebra 1 (2021-2022)
This is the site for the exercice sessions of the first year course Algebra 1, taught by
Gaëtan Chenevier at the ENS on the first term in 2021. Everything is in French.
Exercice sheets (2020-2021)
- TD1 : Ensembles, relations et cardinalité
- TD2 : Exemples de groupes
- TD3 : Quotients et quaternions d'Hamilton
- TD4 : Groupes abéliens
- TD5 : Groupes symétriques et action de groupes
- TD6 : Dévissages
- TD7 : Groupes et géométrie I : les solides de Platon
- TD8 : Groupes et géométrie II : les groupes projectifs
- TD9 : Structure des groupes finis
- TD10 : Arithmétique des anneaux
- TD11 : Modules sur les anneaux principaux
- TD12 : Théorie des représentations I : généralités
- TD13 : Théorie des représentations II : représentations complexes de dimension finie des groupes finies
Corrections-notes
- TD1 : Solutions of some exercices from sheet 1 and a comment on quotient sets.
For more insight around Zagier's proof here .
- TD2 : Solution of some exercices from sheet 2.
- TD3 : Solution of some exercices from sheet 3.
- TD4 : Solution of some exercices from sheet 4.
- TD5 : Solution of some exercices from sheet 5.
- TD6 : Solution of some exercices from sheet 6.
- TD7 : From regular polytopes to finite subgroups of SO(3) and solution of some exercices from sheet 7.
- TD8 : Solution of some exercices from sheet 8.
Extra reading and bibliographie
The following paper (in english) doesn't assume much background about the topic and develops on some ideas concerning angles and curvature of polyhedrons, which we discuss in the sheet 7.
You can find a direct proof of some exceptionnal isomorphisms in this paper (in french) where the most common ones are explicitly constructed. It looks short but is rather condensed.
P. Hall proved a surprising formula concerning p-groups which is well explained in the following short paper. It should be readable only using the course.
Problems
Some problems concerning applications of group theory.
PB1 : Groupes et q-combinatoire.
- PB2 : Polynômes chromatiques et le problème de Polyà.
- PB4 : Induction de représentations et applications.
