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School of Mathematics

# MATH42101 - 2007/2008

General Information
• Title: Representations of Groups
• Unit code: MATH42101
• Credits: 15
• Prerequisites: MATH32001 Group Theory
• Co-requisite units: None
• School responsible: Mathematics
• Members of staff responsible:
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• Course materials will be provided by the lecturer.

## Specification

### Aims

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

### Brief Description of the unit

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.

### Learning Outcomes

On successful completion of this course unit 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.

None.

### Syllabus

1. Informal introduction to matrix representations, permutation representations and G-sets. [3 lectures]
2. Definition and basic properties of complex representations of a finite group. Maschke's Theorem, Characters and character tables. [6]
3. The special cases of: cyclic groups, abelian groups, 1-dimensional representations. [6]
4. Schur's Lemma, orthogonality of characters, the number of irreducibles, the character degree divides the order of the group. [6]
5. Induced representations, Frobenius Reciprocity, double coset formula, methods of calculation. Transitive and 2-transitive permutation representations and their characters. [6]

### Textbooks

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

### Teaching and learning methods

Two lectures and one examples class each week. In additiona students should expect to spend at least seven hours each week on private study for this course unit.

### Assessment

Mid-semester coursework: weighting 20%
End of semester examination: three hours weighting 80%