(Back to module list)
Title
Topics in Abstract Algebra
Code
MM329
Level
3
Credit rating
12 points
Prerequisites
Type
Standard
Aims
To introduce the key notions of algebraic structures with two operations.
To develop in greater depth selected areas of group theory.
To introduce and explore the formal algebraic specification of abstract data types.
To encourage precision in the use of mathematical language, and to develop further the ability to understand and produce proofs in an algebraic context.
Learning outcomes/objectives
By the end of the module the students should be able to:
• understand the basic concepts of group actions and their applications in both algebraic and geometric contexts
• understand the basic concepts of group presentations and use appropriate techniques and reasoning to derive properties of groups defined by generators and relations
• understand the elementary concepts of rings and fields and appreciate the similarities and differences between the these concepts and those of group theory
• comprehend axiomatic presentations of abstract data types and derive and prove their properties.
Content
Group Actions
Definition and examples of group actions, both algebraic and geometric. Orbits and stabilisers; the Orbit-Stabiliser Theorem. Counting orbits; applications.
Group Presentations
Definition and properties of Free groups. Generators and relations. Definitions and examples of finitely-presented groups. Standard presentation of finitely-generated Abelian groups, examples. Derived factor group. Tietze transformations. Presentation of subgroups.
Rings and Fields
Definitions and examples of rings, integral domains and fields. Subrings. Ring morphisms and isomorphisms. Ideals and quotient rings. Polynomial rings. Factorisation in polynomial rings. Unique factorisation domains.
Axiom Systems
Axiomatic presentation of abstract data types, eg, sets, bags, sequences. Consistency, redundancy and derived properties.
Teaching and learning strategies
Six hours of total study time per week of which 3 hours will be timetabled lectures, tutorials and practical classes.
Learning support
Computer software: Maple
Indicative reading:
Armstrong M A Groups and Symmetry 1988 Springer-Verlag
Durbin J Modern Algebra (3rd edition) 1992 Wiley
Fraleigh John A First Course in Abstract Algebra (5th edition) 1994 Addison Wesley.
Gallian Joseph A Contemporary Abstract Algebra (3rd edition) 1994 Lexington.
Johnson D L Presentations of Groups CUP 1990
Kelly J C Abstract Algebra Prentice-Hall 1991
Ledermann W & Weir A J Introduction to Group Theory (2nd edition) 1996 Longman
Loeckx J, Ehrich H-D, Wolf M Specification of Abstract Data Types Wiley 1996
Rotman J J, A First Course in Abstract Algebra 1996 Prentice-Hall
Rotman J J, The Theory of Groups: An Introduction (3rd edition) 1984 Allyn & Bacon
Whitehead C, A guide to Abstract Algebra 1988 Macmillan
The Open University M203 Group Theory Unit 5: Group Actions
Assessment
Coursework: 40%
Examination: 3-hour paper : weight 60%
Brief description of module and/or aims
This module extends the abstract algebra introduced in
MM225 by introducing the topics of rings and fields and developing group theory by considering group actions and group presentations. The application area of the axiom systems and, in particular, the axiomatic presentation of abstract data types is also considered. The emphasis of the module is on understanding of the concepts involved; this will be mainly achieved by investigating examples and generalising from them, but care will be taken to ensure that an appropriate level of rigour and precision is attained.
(Back to module list)