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School of Mathematics

# MATH20201 - 2008/2009

General Information
• Title: Algebraic Structures 1
• Unit code: MATH20201
• Credit rating: 10
• Level: 2
• Pre-requisite units: MATH10101 or MATH10111, MATH10202 or MATH10212
• Co-requisite units:
• School responsible: Mathematics
• Members of staff responsible: Dr C. Eaton
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## Unit specification

### Aims

The programme unit aims to introduce the basic ideas of groups and rings with a good range of examples so that the student has some familiarity with the fundamental concepts of abstract algebra and a good grounding for further study.

### Brief description

This course unit provides an introduction to the main algebraic structures: groups and rings giving the main definitions, some basic results and a wide range of examples. This builds on the study of topics such as properties of the integers, modular arithmetic, and permutations included in MT1101/MT1111. These structures are fundamental concepts in mathematics, particularly in the study of symmetry and of number theory.

### Intended learning outcomes

On completion of this unit successful students will be able to:

• use the basic definitions and properties of groups and rings;
• investigate the basic properties of a good range of examples;
• construct simple proofs of results in abstract algebra.

### Future topics requiring this course unit

This is followed by the semester 2 unit MATH20212 Algebraic Structures 2 and together these provide the basis for a wide range of course units in algebra and related areas at levels 3 and 4. The ideas in this lecture course are also used in analysis, geometry, number theory and topology.

### Syllabus

1. Binary operations. (multiplication tables, associativity, commutativity, associative powers). [1 lecture]
2. Groups. Definitions and examples (groups of numbers, Zn , symmetric groups, groups of matrices). [3]
3. Subgroups. (subgroup criterion, cyclic subgroups, centralizer, centre, order of an element). [4]
4. Cyclic groups. (subgroups of cyclic groups are cyclic, subgroups of finite cyclic groups). [2]
5. Cosets and Lagrange's Theorem. [3]
6. Rings. Definition and examples (rings of numbers, Zn , rings of matrices, quaternions, rings of endomorphisms, group rings, rings of polynomials, subrings). [4]
7. Units, Zero Divisors, Integral Domains and Fields. (cancellation in integral domains, every finite integral domain is a field, Fermat's Little Theorem, characteristic of a field). [3]
8. Homomorphisms and Isomorphisms. [2]

### Learning and teaching processes

• Lectures: Monday 16:00-17:00 Crawford 1; Friday 12:00-13:00 Crawford 1
• Examples classes: Thursdays.
• Family names A-F: 13:00-14:00 Simon 2B;
• Family names G-O: 14:00-15:00 Roscoe A;
• Family names P-Z: 16:00-17:00 Simon 2B.

### Assessment

Coursework; Weighting within unit 20%
2 hours end of semester examination; Weighting within unit 80%

## Arrangements

Online course materials are available for this unit.