Module Identifier MA30110  
Module Title GROUP THEORY  
Academic Year 2004/2005  
Co-ordinator Dr T McDonough  
Semester Semester 2  
Other staff  
Pre-Requisite MA20310  
Course delivery Lecture   19 x 1 hour lectures  
  Seminars / Tutorials   3 x 1 hour examples classes  
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours  100%
Supplementary Exam 2 hours written examination100%

Learning outcomes

On completion of this module, a student should be able to:
A1. determine whether given algebraic systems are groups;
A2. determine whether elements and subsets of a group possess a variety of properties;
A3. state and prove some fundamental theorems, selected from the isomorphism theorems, the orbit-stabilizer theorem and the theorems of Lagrange, Cayley and Sylow;
A4. represent groups as matrix groups, as permutation groups and with generator-relation presentations, and use these representations to compute within the groups;
A5. solve problems in group theory by selecting and applying appropriate theorems and techniques from the general theory.


Brief description

The concept of a group occurs naturally in situations involving symmetry or in which some quantity is being preserved; for example, various letters such as A, S and I possess different numbers of symmetries and rigid motions preserve distance. This module will introduce the notion of a group as an algebraic object defined by a simple set of axioms. Various techniques for describing groups (presentations, matrix and permutation representation) will be studied. The principal structure theorems for finite groups will be described and applied in a variety of group theoretic contexts. Some applications to areas outside of mathematics (e.g. coding, cryptography, crystallography) will be sketched briefly.

Aims

To provide a deeper understanding of the concepts and techniques of abstract algebra, introduced in module MA20310, by focusing on the group concept, starting with an axiomatic development of group theory, establishing a structure theory, mainly in the context of finite groups, and giving brief illustrations of a selection of applications of group theory.

Content

1. Fundamentals: Definitions and examples. Presentations of groups. Elementary consequences of the definitions. Subgroups. cosets. Lagrange'r theorem.
2. Basic structure theory: Normal subgroups and factor groups. Direct products. Homomorphisms. The isomorphism theorems. Automorphism groups.
3. Permutation groups: Symmetric groups. Cycle decomposition. Regular representation. Cayley's theorem. Orbits. Stabilizers. The orbit-stabilizer theorem.
4. Local structure theory: p-subgroups. The Sylow theorems. Classifying groups of small order.
5. Global structure theory: Classification of finite Abelian groups. The Jordan-Holder theorem and its consequences.

Reading Lists

Books
** Recommended Text
J F Humphreys (2001) A Course in Group Theory Oxford University Press 0198534590
** Supplementary Text
D W Farmer (1996) Groups and Symmetry 6th Ed. American Mathematical Society 0821804502
J B Fraleigh (2003) A First Course in Abstract Algebra 7th Ed. Addison-Wesley 0321156080
C R Jordan and D A Jordan (1994) Group Theory Edward Arnold 034061045X
J R Durbin (2000) Modern algebra : an introduction 4th Ed. John Wiley and Sons, Inc. 0471321478
D A R Wallace (1998) Groups Rings and Fields Springer 3540761772
B Baumslag and B Chandler (1968) Theory and Problems of Group Theory McGraw-Hill 0070041245

Notes

This module is at CQFW Level 6