5.00 credits
30.0 h + 15.0 h
Q1
This biannual learning unit is being organized in 2021-2022
Teacher(s)
Caprace Pierre-Emmanuel;
Language
French
Prerequisites
LMAT1131 - linear algebra (first year Bachelor of Mathematical Sciences) or 'equivalent' course.
LMAT1231 - multilinear algebra and group theory (second year Bachelor of Mathematical Sciences) or equivalent course.
LMAT1331 - commutative algebra (third year Bachelor of Mathematical Sciences) or equivalent course.
LMAT1231 - multilinear algebra and group theory (second year Bachelor of Mathematical Sciences) or equivalent course.
LMAT1331 - commutative algebra (third year Bachelor of Mathematical Sciences) or equivalent course.
Main themes
Group theory provides a formal framework for studying the notion of symmetry.
The ubiquity of groups in mathematics and physics stems from the prominent role played by the notion of symmetry in science. The course will provide a broad introduction to the theory of groups, both finite and infinite, using explicit examples.
The following topics will be covered :
- Basic examples and constructions
- Permutation groups, linear and projective groups
- Free groups and group presentations
- Resolvable and nilpotent groups, links with Galois theory
- Abelian groups and Pontryagin duality
- Construction of simple, finite and infinite groups
- Burnside problem on torsion groups
- Groups acting on trees, hyperbolic groups
- Representations, characters of finite groups and applications
From one year to the next, some aspects will be developed more than others.
The ubiquity of groups in mathematics and physics stems from the prominent role played by the notion of symmetry in science. The course will provide a broad introduction to the theory of groups, both finite and infinite, using explicit examples.
The following topics will be covered :
- Basic examples and constructions
- Permutation groups, linear and projective groups
- Free groups and group presentations
- Resolvable and nilpotent groups, links with Galois theory
- Abelian groups and Pontryagin duality
- Construction of simple, finite and infinite groups
- Burnside problem on torsion groups
- Groups acting on trees, hyperbolic groups
- Representations, characters of finite groups and applications
From one year to the next, some aspects will be developed more than others.
Learning outcomes
At the end of this learning unit, the student is able to : | |
| 1 | At the end of this activity, the student will have progressed in his/her ability to : (a) Know and understand a fundamental foundation of mathematics. In particular, he/she will have developed the ability to : i. Recognize the fundamental concepts of important current mathematical theories. ii. Establish the main links between these theories. (b) Demonstrate abstraction, reasoning and critical thinking skills. In particular, the student will have developed the ability to : i. Identify unifying aspects of different situations and experiences. ii. Reason within the framework of the axiomatic method. iii. Construct and write a demonstration in an independent, clear and rigorous manner. (c) analyze a mathematical problem and propose adequate tools to study it independently. |
Content
This course aims at introducing some of the fundamental concepts from the theory of groups. A special emphasis is put on finitely generated infinite groups, and their study by geometric methods.
The following themes will be discussed, starting from concrete examples.
The following themes will be discussed, starting from concrete examples.
- Abelian, nilpotent and soluble groups.
- Free groups and groups acting on trees, theorem of Nielsen-Schrier.
- Structure of the autormorphism group of a vector space, and the orthogonal groups
- Linear groups and residual finiteness.
- Word problem and decidability
Teaching methods
The course is taught through lectures and practical exercises. In the practical exercise sessions, students will be asked to make suggestions and formulate ideas in order to further the course on the basis of their prior knowledge.
Evaluation methods
The evaluation for this course consists of :
- continuous evaluation, including projects during the quadrimester (50% of the final grade),
- an oral examination during the exam session (50% of the final grade).
Other information
This biennial course is taught during the academic year 2021-2022, but not during the year 2022-2023.
Online resources
Bibliography
C. Drutu and M. Kapovich, Geometric Group Theory. American Mathematical Society Colloquium Publications 63, 2018.
P. de la Harpe, Topics in Geometric Group Theory. Chicago Lectures in Mathematics, 2000.
J. Meier, Groups, graphs and trees. An introduction to the geometry of infinite groups. London Mathematical Society Student Texts 73, Cambridge UP, 2008.
D. Robinson, A course in the theory of groups. (Second edition). Graduate Texts in Mathematics, Springer, 1996.
J. Rotman, An introduction to the theory of groups. Graduate Texts in Mathematics, Springer, 1995.
J.-P. Serre, Arbres, amalgames, SL2. Astérisque, No. 46. Société Mathématique de France, Paris, 1977.
P. de la Harpe, Topics in Geometric Group Theory. Chicago Lectures in Mathematics, 2000.
J. Meier, Groups, graphs and trees. An introduction to the geometry of infinite groups. London Mathematical Society Student Texts 73, Cambridge UP, 2008.
D. Robinson, A course in the theory of groups. (Second edition). Graduate Texts in Mathematics, Springer, 1996.
J. Rotman, An introduction to the theory of groups. Graduate Texts in Mathematics, Springer, 1995.
J.-P. Serre, Arbres, amalgames, SL2. Astérisque, No. 46. Société Mathématique de France, Paris, 1977.
Teaching materials
- matériel sur moodle
Faculty or entity
MATH
