Group Theory
| Code | School | Level | Credits | Semesters |
| MATH4089 | Mathematical Sciences | 4 | 20 | Autumn UK |
- Code
- MATH4089
- School
- Mathematical Sciences
- Level
- 4
- Credits
- 20
- Semesters
- Autumn UK
Summary
This course starts with basic examples of groups and their actions on a plane. The material is focused on illustrating how a concept of a symmetry via its algebraic interpretation gives rise to numerous applications in a wide range of subjects, including combinatorics, probability, number theory, harmonic analysis and mathematical physics. Topics for this course include:
• Transformations of a plane: rotation and reflections. Groups O(2,R), SO(2,R) and their finite subgroups.
• Basic concepts of group theory: subgroups, co-sets, quotient groups. Lagrange's theorem.
• Group homomorphisms and their properties.
• Basic results on actions of finite groups.
• Adjoint action. Applications of symmetric groups.
• Sylow Theorems and simplicity tests.
• Finitely generated abelian groups.
• Characters of finite abelian groups.
Target Students
Single and Joint Honours students from the School of Mathematical Sciences. Available to Postgraduate Taught students and Postgraduate Research students.
Classes
- One 2-hour lecture each week for 11 weeks
- Two 1-hour lectures each week for 11 weeks
One two-hour class and two one-hour classes per week timetabled centrally, some of which may be used for lectures or workshops.
Assessment
- 100% Exam 1 (3-hour)
Assessed by end of autumn semester
Educational Aims
The course develops the basic theory of appropriate mathematical notions arising from Geometry. In particular, it helps to see how geometric methods and constructions can be exported to other areas of Mathematics. The course starts with transformations of a plane and then develops the basic theory of finite groups and finitely generated Abelian groups, deepening understanding of the abstract ideas and logical arguments involved with emphasis on geometric meaning of these fundamental constructions.Learning Outcomes
L1 - introduce the main concepts and theorems of group theory;
L2 - understand the fundamental results on group theory;
L3 - apply group theoretic reasoning to group actions, symmetry in space;
L4 - relate structural results for groups with vector spaces and geometry;
L5 - prove results in group theory using abstract and rigorous reasoning.