MATH40012 Representation of GroupsThis is former 411/MA4010 SEMESTER: First CONTACT: Dr Charles Eaton (M/Q2) CREDIT RATING: 15
 Aims: To introduce students to representations of groups over the field of complex numbers. Intended Learning Outcomes: On successful completion of the course students will: Know the basic properties of complex representations of finite groups and be able to use them in examples. Understand the relationship between a representation and its character. Know the basic properties of characters and use them in examples. Know the basic properties of a character table and be able to calculate character tables for certain small groups. Pre-requisites: 212, 252, 312 (ex-UMIST) Dependent Courses: None Course Description: In the second and third year modules on group theory we have seen that abstract groups are quite complicated objects. One of the most fruitful approaches to studying these objects is to embed them into groups of matrices (to "represent" the elements of an abstract group by matrices). The advantage of this approach lies in the fact that matrices are concrete objects, and explicit calculations can easily be performed. Even more importantly, the powerful methods of linear algebra can be applied to matrices. The course is devoted to representations of finite groups by matrices with entries in the field of complex numbers. Teaching Mode: 2 Lectures per week 1 Tutorial per week Private Study: 5 hours per week Recommended Texts: J P Serre, Linear Representations of Finite Groups, GMT 42, Springer-Verlag G James and M Liebeck, Representations and Characters of Groups, CUP, 1993 Assessment Methods: Coursework: 20% Coursework Mode: Problem sheets.  Deadlines in Weeks 7 and 10. Examination: 80% A 2 hour examination at the end of the First Semester.
 No of lectures Syllabus 3 Informal introduction to matrix representations, permutation representations and G-sets. 6 Definition and basic properties of complex representations of a finite group. Maschke’s Theorem, Characters and character tables. 6 The special cases of: cyclic groups, abelian groups, 1-dimensional representations. 6 Schur’s Lemma, orthogonality of characters, the number of irreducibles, the character degree divides the order of the group. 6 Induced representations, Frobenius Reciprocity, double coset formula, methods of calculation.  Transitive and 2-transitive permutation representations and their characters.

Last revised August, 2006