|| MA30110 |
|| GROUP THEORY |
|| 2003/2004 |
|| Dr T McDonough |
|| Semester 2 |
|| Dr T McDonough |
|| MA20310 |
| Course delivery
|| Lecture || 19 x 1 hour lectures |
|| Seminars / Tutorials || 3 x 1 hour examples classes |
|Assessment Type||Assessment Length/Details||Proportion|
|Semester Exam||2 Hours ||100%|
|Supplementary Exam|| 2 hours written examination||100%|
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 a selection of fundamental theorems, including 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.
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.
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.
Fundamentals: Definitions and examples. Presentations of groups. Elementary consequences of the definitions. Subgroups. cosets. Lagrange?s theorem.
Basic structure theory: Normal subgroups and factor groups. Direct products. Homomorphisms. The isomorphism theorems. Automorphism groups.
Permutation groups: Symmetric groups. Cycle decomposition. Regular representation. Cayley's theorem. Orbits. Stabilizers. The orbit-stabilizer theorem.
Local structure theory: p-subgroups. The Sylow theorems. Classifying groups of small order.
Global structure theory: Classification of finite Abelian groups. The Jordan-Holder theorem and its consequences.
Other applications: Brief sketches of a selection of applications of group theory, e.g. permutation decoding, RSA and elliptic curve cryptosystems, crystallography, space-time and the Lorentz group.
J F Humphreys (2001) A Course in Group Theory
Oxford University Press 0198534590
** Supplementary Text
J R Durbin (2000) Modern algebra : an introduction
4th Ed. John Wiley and Sons, Inc. 0471321478
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
D A R Wallace (1998) Groups Rings and Fields
B Baumslag and B Chandler (1968) Theory and Problems of Group Theory
This module is at CQFW Level 6