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# MMath Mathematics and Statistics / Course details

Year of entry: 2021

## Course unit details:Representation and Characters of Groups

Unit code MATH62062 15 FHEQ level 7 – master's degree or fourth year of an integrated master's degree Semester 1 Department of Mathematics No

### Overview

In the second and third year course units 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.

### Pre/co-requisites

Students are not permitted to take, for credit, MATH42061  in an undergraduate programme and then MATH62061 in a postgraduate programme at the University of Manchester, as the courses are identical.

### Aims

To introduce students to representations of groups over the field of complex numbers.

### Learning outcomes

On successful completion of this course unit students will be able to:

• Prove elementary properties of representations and characters.
• Prove Maschke's Theorem, Schur's Lemma and related results.
• State, prove and apply the orthogonality of characters.
• Construct the induced representation and prove Frobenius reciprocity.
• Calculate the character tables of certain groups.
• Apply representation theory to various problems on the structure of groups and G-sets.

### Syllabus

• Informal introduction to matrix representations, permutation representations and G-sets. [3 lectures]
• 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. [6]

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Mid-semester coursework: weighting 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• B. Steinberg, Representation Theory of Finite Groups, Springer Verlag 2012
• J. P. Serre, Linear Representations of Finite Groups, GMT 42, Springer-Verlag
• G. James and M. Liebeck, Representations and Characters of Groups, CUP, 1993

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 117

### Teaching staff

Staff member Role
Peter Symonds Unit coordinator