Fergal Gallagher

Finite Group Algebras and their Applications in Manufacturing Engineering

A research project by Fergal Gallagher   

Supervisor: Dr. Leo Creedon, lecturer in Mathematics at IT Sligo.
School of Engineering, Institute of Technology Sligo

Introduction: A Group Algebra is an algebraic object consisting of the elements of a group with co-efficients from a field. An example of a Group Algebra is a 3-dimensional Vector Space with real number co-efficients. Another example is the polynomial ax2+bx+c, where a, b and c are integers. A Finite Group Algebra consists of a group with co-efficients from a finite field. Group Algebras have many applications in manufacturing including communications and digital technology.

Aims and Objectives: The aim of this project is to produce a list and description (an atlas) of the smaller unit groups of finite group algebras. A similar project creating an atlas of finite groups has been completed and has proved to be fundamental to our understanding of groups.

Methodology: The process will begin with discovering and listing the size and abstract structure of the unit groups of small finite cyclic and abelian groups over a finite field. This process will continue with the study of the unit groups and zero divisors of the dihedral groups D6 and D10 over a finite field. This will be done by both theoretical and computer modelling techniques.

Results: My aim is to describe the unit group of FG, where F is any finite field and G is any group of order less than 8 as well as many classes of groups of much larger order. The results of this research will will form a fundamental part of the atlas of finite group algebras.

Conclusion: This is a very exciting area of research in a relatively new area of mathematics with a range of applications including coding and digital communications.