| Unit name | Number Theory and Group Theory |
|---|---|
| Unit code | MATH11511 |
| Credit points | 10 |
| Level of study | C/4 |
| Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |
| Unit director | Dr. Walling |
| Open unit status | Not open |
| Pre-requisites |
A good A level pass in Mathematics or equivalent. |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
The unit starts with some basic number theory including prime numbers, common factors, the division algorithm and Euclid's algorithm, the Fundamental Theorem of Arithmetic, and congruence of integers. This material, in addition to being of interest in its own right, is a good setting for the students to meet and practise clear logical thinking and various methods of proof.
Then there is an introduction to group theory which will last till the end of the unit. In the past, certain systems studied in various parts of mathematics have turned out to have common features, and these have been formalised into the definition of a group. Some of the earliest examples arose in connection with the solution of polynomial equations by formulae, and involved what we would now call groups of permutations. Other examples arise in trying to pin down mathematically what it means to say that a geometrical figure is symmetric and to quantify just how symmetric it is. It makes sense to study in one go all the systems which have the same general features. We shall start from the formal definition of a group and derive important general results from it using careful mathematical reasoning, but throughout there will be an emphasis on particular examples in which calculations can be performed relatively easily.
Aims:
This unit aims to develop students' ability to think and express themselves in a clear logical fashion, and to introduce basic material on number theory and group theory.
Syllabus
Number Theory: Integers; divisibility; common factors; the division algorithm and Euclid's algorithm; the equation ax + by = c; prime numbers and the Fundamental Theorem of Arithmetic; congruence of integers; Fermat's Little Theorem; solution of linear congruences. [8 lectures]
Group Theory: Definitions and examples. [3 lectures]
Subgroups. [1 lecture]
Order of an element. [1 lecture]
Cyclic groups. [1 lecture]
Direct products. [2 lectures]
Isomorphic groups. [1 lecture]
Lagrange's theorem and some applications. [2 lectures]
Groups of permutations. [3 lectures]
Relation to Other Units
This unit is the foundation for Algebra 2 and other algebra and number theory units in later years.
Relation to Other Units
This unit is the foundation for Algebra 2 and other algebra and number theory units in later years.
After taking this unit students should: - Be able to understand and write clear mathematical statements and proofs; - Be proficient in using Euclid's algorithm and manipulating congruences, and understand the basic properties of prime numbers; - Have acquired facility in working with various specific examples of groups; - Be able to solve standard types of problems in elementary number theory and group theory; - Understand and be able to apply the basic concepts and results presented throughout the unit.
Transferable Skills:
The ability to express intuitive ideas in a precise mathematical fashion and to produce clear logical arguments.
The course will be based on lectures and (for first year students) small group tutorials. Homework exercises will be marked by tutors or by the lecturer and model solutions will be provided. Notes will be provided by the lecturer, but access to the suggested books (especially the recommended book on group theory) may be helpful .
100% examination.
The following is recommended:
The following may also be useful:
