Topics in Group Theory
This is an original introductory course on Group Theory based on examples. Group Theory is a rigorous way to use the concept of symmetry in Mathematics with wide range of applications. The course addresses a diverse audience (of undergraduate students) with diverse background. We introduce basic concepts and methods of group actions, using motivation from Geometry and Number theory. We apply those results to studying structure of finite groups, and illustrate them with examples. This course is aiming at helping the students to see how concept of symmetry connects different areas of Mathematics, and to develop their individual research interests.
Lecturer
Date
7th March ~ 31st May, 2024
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Thursday | 13:30 - 15:05 | A3-1-103 | ZOOM 11 | 435 529 7909 | BIMSA |
| Friday | 09:50 - 11:25 | A3-1-103 | ZOOM 11 | 435 529 7909 | BIMSA |
Prerequisite
basic Algebra
Syllabus
1. Definition of a group and main examples. Maps X --> X and permutations. Linear transformations and GL(n,R). Additive and multiplicative groups modulo natural n.
2. Co-sets and normal subsgroups. Quotient groups. Examples: (R,+)/(Z,+); S_n/A_n; O(n,R)/SO(n,R); (Z,+)/(nZ,+)
3. Orders of group elements. Cyclic (sub)groups: finite and infinite. Fundamental theorem on cyclic groups.
4. Group homomorphisms. Isomorphism theorems. Direct products. Chinese Remainder Theorem.
5. Definition of group action. Examples. Action on co-sets. Cayley's Theorem.
6. Adjoint action. Conjugation action on subgroups.
7. The symmetric group. Cycle decomposition and Young Tableaux. Partition number and Ramanujan's congruences. Abelian p-groups..
8. Automorphism groups. Isometry groups.
9. Semidirect products. Sylow Theorems.
10. Finitely generated Abelian groups. Smith normal form. Fundamental Theorem.
11. Group extensions. Central extensions. The first and the second cohomology groups.
2. Co-sets and normal subsgroups. Quotient groups. Examples: (R,+)/(Z,+); S_n/A_n; O(n,R)/SO(n,R); (Z,+)/(nZ,+)
3. Orders of group elements. Cyclic (sub)groups: finite and infinite. Fundamental theorem on cyclic groups.
4. Group homomorphisms. Isomorphism theorems. Direct products. Chinese Remainder Theorem.
5. Definition of group action. Examples. Action on co-sets. Cayley's Theorem.
6. Adjoint action. Conjugation action on subgroups.
7. The symmetric group. Cycle decomposition and Young Tableaux. Partition number and Ramanujan's congruences. Abelian p-groups..
8. Automorphism groups. Isometry groups.
9. Semidirect products. Sylow Theorems.
10. Finitely generated Abelian groups. Smith normal form. Fundamental Theorem.
11. Group extensions. Central extensions. The first and the second cohomology groups.
Reference
1) Standard textbooks covering finite groups, for example:
W.Ledermann, Introduction to Group Theory, Longman, 1973
J.F.Humphreys, A course in group theory, Oxford, 1996
M.A.Armstrong, Groups and Symmetry, Springer, 1988
2) Further reading:
J.Carrell, Groups, Matrices and Vector Spaces: A Group Theoretic Approach to Linear Algebra, Springer, 2017
E.Artin, Geometric Algebra, Princeton University Press, 1957
W.Ledermann, Introduction to Group Theory, Longman, 1973
J.F.Humphreys, A course in group theory, Oxford, 1996
M.A.Armstrong, Groups and Symmetry, Springer, 1988
2) Further reading:
J.Carrell, Groups, Matrices and Vector Spaces: A Group Theoretic Approach to Linear Algebra, Springer, 2017
E.Artin, Geometric Algebra, Princeton University Press, 1957
Audience
Undergraduate
, Advanced Undergraduate
, Graduate
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Sergey Oblezin received his PhD at Moscow Institute of Physics and Technology in 2004. Education in Moscow and work experience at the Alikhanov Institute for Theoretical and Experimental Physics shaped his intra-disciplinary vision in mathematics, based on a unique and mutually transformative synthesis of quantum physics and mathematics. At early stage, his research achievements were recognized by several awards including two Russian Federation President Fellowships for young mathematicians (in 2007-2008 and 2008-2009). In 2009-2012, Sergey's research was awarded by the Pierre Deligne Prize (supported by P.Deligne's Balzan Prize, 2004). In 2013-17 Sergey's project "Topological field theories, Baxter operators and the Langlands programme" was supported by the Established Career EPSRC grant (UK). During 2015-2023, Sergey was an Associate Professor in Geometry at the University of Nottingham (UK), before taking his current full-time Professor position at BIMSA in 2024. Sergey Oblezin is working on a long term research project devoted to transferring and developing methods and constructions of quantum physics to the Langlands Program. His research interests include representation theory, harmonic analysis and their interactions with number theory and mathematical physics.