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2010/1 Module Catalogue
 Module Code: MAT2006 Module Title: GROUPS AND SYMMETRY
Module Provider: Mathematics Short Name: MS215
Level: HE2 Module Co-ordinator: FISHER D Dr (Maths)
Number of credits: 15 Number of ECTS credits: 7.5
 
Module Availability
Semester 1
Assessment Pattern

                        Unit(s) of Assessment

 
                        Weighting Towards Module Mark( %)

 

Coursework: two class tests with preparatory assignments

 

25

 

  

Examination:  two hours unseen

 

75

 

  

Qualifying Condition(s) 

 

A weighted aggregate mark of 40% is required to pass this module.

 

 

  
Module Overview
      The module combines algebraic structure theory with applications to symmetries.
Prerequisites/Co-requisites
MAT1016 Linear Algebra
Module Aims
The course aims to introduce the axiomatic approach to group theory, to show that group theory is the natural language of symmetry, and to cover a range of fundamental pure and applied topics in the theory of groups such as subgroups, morphisms, generators, quotient and product groups, cosets, conjugacy and actions.     
Learning Outcomes
At the end of the module the student should have a basic understanding of the abstract concept of a group and its role in analysing symmetry; a familiarity with the basic groups such as dihedral, symmetric, cyclic and orthogonal groups, and an understanding of the analysis and structure of groups.
Module Content
The contents of the lectures will include:
 
Introduction: symmetry plays a central role in mathematics as well as in nature, art and science. The language of symmetry is group theory. The module will develop and study the axiomatic theory of groups; including such issues as group actions, subgroups, generators, cosets, homomorphisms, conjugacy, commutativity, quotient and product constructions. Examples of groups to be considered are cyclic groups, dihedral groups, symmetric groups and orthogonal groups.
 
Subgroups: the concept of a subgroup will be introduced and special subgroups such as normal subgroups and the centre of a group, as well as the concept of a generator and the role of Lagrange's theorem, will be presented.
 
Quotient and product constructions: direct products and quotient groups will be introduced, as well as the role of cosets.
 
Morphisms: the concepts of isomorphism, homomorphism and automorphism will be introduced and applied to the study of groups.
 
Groups: examples of groups to be studied are the dihedral groups, reflection groups, cyclic groups, permutation groups and orthogonal groups.
Methods of Teaching/Learning

Teaching is by lectures and tutorials, 3 hours per week for 11 weeks.

 
Selected Texts/Journals
M.A. Armstrong, Groups and Symmetry, Springer-Verlag, (1988), ISBN 0387966757
 
J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley (2003), ISBN 0321156080 (or other edition)
 
R.P. Burn, Groups: A Path to Geometry, Cambridge University Press, (1985).
 
J.A. Green, Sets and Groups, Routledge & Kegan Paul, (1965).
Last Updated
September 10