MATH20201

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]

2. Groups. Definitions and examples (groups of numbers, Z_n , 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. Costs and Lagrange's Theorem. [3]

6. Rings. Definition and examples (rings of numbers, Z_n , 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]

Textbooks

John B. Fraleigh, A First Course in Abstract Algebra, Addidon-Wesley

Learning and teaching processes

Two lectures and one examples class each week

Assessment

Coursework Weighting within unit 20%

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

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Arrangements