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Number Theory and Group Theory (MATH 11511)

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Administrative Information

  1. Unit number and title: MATH 11511 Number Theory and Group Theory
  2. Level: C/4 (Honours)
  3. Credit point value: 10 credit points
  4. Year: 12/13
  5. First Given in this form: 1999/2000
  6. Unit Organiser: Lynne Walling
  7. Lecturer: Dr. Sean Prendeville
  8. Teaching block: 1
  9. Prerequisites: An A in A level Mathematics or equivalent.

Unit 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.

General Description of the Unit

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.

Relation to Other Units

This unit is the foundation for Algebra 2 and other algebra and number theory units in later years.

Teaching Methods

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 .

Learning Objectives

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.

Assessment Methods

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).

Award of Credit Points

To be awarded the credit points for this unit you must normally pass the unit, i.e. you must achieve an assessment mark of at least 40.

The assessment mark is calculated as described in the Assessment section above. Details of the university's common criteria for the award of credit points are set out in the Regulations and Code of Practice for Taught Programmes at http://www.bristol.ac.uk/esu/assessment/codeonline.html

Note that for this unit:

  • first year students are expected to attend all the relevant tutorials,
  • all students are expected to hand in attempts to the weekly exercises set.

Transferable Skills

The ability to express intuitive ideas in a precise mathematical fashion and to produce clear logical arguments.

Texts

The following is recommended:

"Groups" by C. R. Jordan and D. A. Jordan, originally published by Edward Arnold in 1994, reprinted by Newnes (Elsevier) in 2001, 2003, 2004; ISBN 0-340-61045-x.

The following may also be useful:

"How to prove it" by D. J. Velleman.

"An Introduction to mathematical reasoning" by PJ. Eccles (CUP)

"Numbers, goups and codes" 2nd Edition by JF Humphreys and MY Prest (CUP)

"Adventures in Group Theory" by D. Joyner (John Hopkins)

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]