MATH32001/42001 - 2007/2008
- Title: Group Theory
- Unit code: MATH32001/42001
- Credits: 10 (MATH32001), 15 (MATH42001)
- Prerequisites: MATH20212 Algebraic Structures 2
- Co-requisite units: None
- School responsible: Mathematics
- Member of staff responsible:
This lecture course aims to introduce students to some more sophisticated concepts and results of group theory as an essential part of general mathematical culture and as a basis for further study of more advanced mathematics.
Brief Description of the unit
The ideal aim of Group Theory is the classification of all groups (up to isomorphism). It will be shown that this goal can be achieved for finitely generated abelian groups. In general, however, there is no hope of a similar result as the situation is far too complex, even for finite groups. Still, since groups are of great importance for the whole of mathematics, there is a highly developed theory of outstanding beauty. It takes just three simple axioms to define a group, and it is fascinating how much can be deduced from so little. The course is devoted to some of the basic concepts and results of Group Theory.
On successful completion of this course unit students will have acquired
- a sound understanding of the classification of finitely generated abelian groups,
- knowledge of some fundamental results and techniques from the theory of finite groups,
- knowledge of group actions on sets, simple groups, Sylow's theorems and various applications of Sylow's theorems.
Future topics requiring this course unit
MATH42101 Representations of Groups, MATH42122 Galois Theory.
- Revision of basic notions (subgroups and factor groups, homomorphisms and isomorphisms), generating sets, commutator subgroups. [2 lectures]
- Abelian groups, the Fundamental Theorem on finitely generated abelian groups. 
- The Isomorphism Theorems. 
- Simple groups, the simplicity of the alternating groups. 
- Composition series, the Jordan-Hölder Theorem. 
- Group actions on sets, orbits, stabilizers, the number of elements in an orbit, Burnside's formula for the number of orbits, conjugation actions, centralizers and normalizers. 
- Sylow's Theorems, groups of order pq, pqr. 
- John B Fraleigh, A First Course in Abstract Algebra, (5th edition), 1967, Addison-Wesley.
Teaching and learning methods
Two lectures and one examples class each week.
Four hours private study each week (seven hours for MATH42001).