CM351A Representation Theory of Finite Groups 6CCM351A Representation Theory of Finite Groups
Office hours: Mondays 2-3pm, office 405
Teaching arrangements: Three hours of lectures per week (in the second semester); some lectures can be used as tutorials and question sessions.Prerequisites: Introduction to Abstract Algebra, Linear Algebra, Groups and Symmetries (or equivalents).
Assignments: There will be some assignments for practice. These will be discussed in tutorials.
Aims and objectives: The aim of this module is to develop the basic theory of linear representations and characters of finite groups over the complex numbers.
Syllabus: The basic definitions and standard properties of linear representations of finite groups over the complex numbers (in particular Schur's lemma and Maschke's theorem). The relation between representations and characters, the orthogonality relations and other fundamental properties of characters and character tables. Application of the above results to performing explicit calculations for groups of small order.
One of the following more advanced topics will be covered: induction and restriction of representations and characters, algebraic integers and their applications to characters of finite groups, or representations over the real numbers
REVISION NOTES on groups and linear algebra from course at Univ of Edinburgh: Gordon revision notes (pdf) (handed out in first lecture)
FIRST TUTORIAL: Friday February 3, 12pm
SECOND TUTORIAL: Monday February 20, 2nd hour
There will be no lecture Friday February 17.
PROBLEM SHEET 1: pdf file
FULL SOLUTIONS 1: pdf file
LEVEL 7 BONUS PROBLEM: pdf file
PROBLEM SHEET 2: pdf file
DEFINITIONS SHEET (pages 1 and 2, updated 6/2): pdf file
SKELETON NOTES part 1: pdf file
Books, course material:Walter Ledermann: "Introduction to group characters",
James and Liebeck: "Representations and Characters of Groups",
The first half of: J-P. Serre, "Linear representations of finite groups".
Previous years' exams:
May 2010: pdf file
May 2009: pdf file
Pointers to optional supplementary reading:
Notes from Cambridge: Teleman lectures (pdf)
See particularly Sections 1,2,3,4 (excluding prop's 2.2, 4.7, 4.10), Section 5 from (5.6) onwards, Section 6 (excluding 6.3,6.5,6.8), Section 8 and 9 (up to but not including (9.6)), Sections 10, 14, 15. [However these notes miss out the Cauchy-Frobenius Lemma and some of the results on permutation representations we will also cover.]Notes on group actions from Purdue's Abstract Algebra Course: Go to the Lecture notes page and see particularly,