| 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. Jordan |
| 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 |
A rigorous development of basic number theory and group theory. Topics in number theory include prime numbers, common factors, division algorithm and Euclid's algorithmm, the fundamental theorem of arithmetic, and congruence of integers. Topics in group theory include definition of an abstract group, basic examples such as symmetry groups and multiplicative groups of integers to a modulus, notions of order, subgroups, Abelian and cyclic groups, direct products, Lagrange's theorem, permutation groups.
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.
After taking this unit students should:
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;
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. Duplicated notes will be provided by the lecturer, but access to the suggested books (especially the recommended book on group theory) may be helpful.
The final mark for Number Theory and Group Theory is calculated as follows:
100% from a 1½-hour examination in April More information is given below.
April Examination The examination in April consists of one 1½-hour paper, in two sections.
Section A contains 5 short questions, ALL of which should be attempted. Section A contributes 40% of the mark for this paper. Section B has 3 longer questions; you should attempt TWO. If you attempt more than two, your best two answers in Section B will be used for assessment. Section B contributes 60% to the mark for this paper. Calculators may NOT be used.
September examinations If you fail Number Theory and Group Theory (or any other unit in the Science Faculty), you may be required to resit in the first half of September. Your departmental or Faculty handbook explains the conditions under which resits may be allowed. The September examinations have the same format as the April examination (given above).
The following is recommended:
The following may also be useful:
