# module specification

## MA5030 - Discrete Mathematics and Group Theory (2017/18)

Module specification Module approved to run in 2017/18
Module title Discrete Mathematics and Group Theory
Module level Intermediate (05)
Credit rating for module 30
School School of Computing and Digital Media
Total study hours 300

 90 hours Scheduled learning & teaching activities 210 hours Guided independent study
Assessment components
Type Weighting Qualifying mark Description
In-Course Test 25%   Test (1 hour)
Coursework 25%   Coursework
Unseen Examination 50%   Exam 2 hours
Running in 2017/18
Period Campus Day Time Module Leader
Year North Monday Afternoon

## Module summary

The topics covered in the first term of this module is to introduce formal inductive and recursive structure on the natural numbers. This structure underlies many aspects of program design and validation, and formal methods. An introduction to combinatorics and the generetaing functions are designed to enhance the students algorithmic tool set.

The topics covered in the second term part of this module is to introduce students to the abstract algebraic structures of groups, which arise from the ideas of symmetries and of vector and matrix calculus repectively. These two primary examples of algebraic structures have applications across science and engineering, and also provide a firm foundation of necessary basic algebraic notions for the student to further their study mathematical study.

## Prior learning requirements

Successful completion of Certificate Level of any BSc in Mathematics area

## Module aims

The aims of the  module are: to equip students with a sound understanding of the concepts of recursion and induction in mathematics and of their practical application to a number of structures; The aim of the second part of the module is  to introduce students to the abstract algebraic structures of groups s, which arise from the ideas of symmetries . This primary example of algebraic structures has applications across science and engineering, and also provides a firm foundation of necessary basic algebraic notions for the student to further their study in mathematics.

## Syllabus

• The Natural Numbers: Induction. Axioms for addition and multiplication. Recursively defined functions. The abstract data type (ADT) for the Natural numbers
• Combinatorics: Functions and Principles of Counting.  Functions, Words and Selections. Binomial numbers and  the Binomial Theorem. Ordered and unordered selections. with or without repetition. Partitions , Distributions and the  Multinomial Numbers.   Generating Functions: Definition and operations. Using generating functions to solve recursions. Difference Equations.
• Revision of notions of binary operation, associativity.
• Formal definition of a group. Worked examples: symmetries of a regular polygon.
• Generators, orders. Abelian groups, cyclic groups, free groups, relations in groups.
• Subgroups. Cosets and Theorem of Lagrange.  Normal subgroups; quotient groups.
• Todd-Coxeter method for calculating order of a group and finding coset transversals
• Homomorphisms and isomorphisms. Kernel and image; first isomorphism theorem.
• Permuation groups: symmetric and alternating groups. Cayley's theorem.
• Groups of matrices

## Learning and teaching

The module will be taught by a mixture of lectures, workshops and self study practical exercises. The lectures (22 hours) will be used to introduce the various concepts and principles of the module and their strengths in applications. Lectures will be followed by workshops (90 hours).

The workshops will afford students the opportunity to work in small groups on exercises related to previously taught material. The students will be able to present previously completed exercises for comment from the lecturer and other students.  In this class time students will also be encouraged to explore and experiment with the concepts and techniques to encourage their own sense of mathematical creativity.

Students will be expected to spend time on unsupervised work, for example, private study of problem sheets and in the preparation of  their individual coursework (210 hours).  A framework will be put in place to encourage disciplined learning through student self-awareness of progress in volume of work, understanding, attendance and punctuality.

In addition to standard VLE presence there will be links available for further readings and discussion groups.

## Learning outcomes

 LO1: Manipulate recursive definitions of functions and use inductive proofs on the natural numbers and abstract data types and express these proofs carefully as logical arguments. LO2: Derive definitions for ordered/unordered selections using subset argument, understand combinations/ permutations/words as number of special functions LO3: Understand the concept of generating functions, their operations and application to solving recursions LO4: Understand the Todd-Coxeter method and apply it to gain further insights into the structures of groups. LO5: appreciate notions and theory of subgroups and quotient groups  and apply them correctly to well defined problems and  construct homomorphisms between groups and determine their properties

## Assessment strategy

There will be one progression test covering LO1-3 which will give students opportunity to demonstrate their understanding of selection of topics. The Coursework will assess L04. The final assessment will be an exam where students will tested on LO1-5.

## Bibliography

Hein, J L, (2003), Discrete Mathematics (2nd ed), Jones and Bartlett. (512.5)
Mattson, H F(1993), Discrete Mathematics with Applications, John Wiley & Sons.
Gorman, S.P. et al.;  Lecture Notes in Groups.  University of North London.
Jordan, C and Jordan, D(1994),  Groups.  E. Arnold.
Fraleigh, J.B. (1976);  A first course in abstract algebra.  Addison-Wesley