Algebraic Structures 1

 Unit code: MATH20201 Credit Rating: 10 Unit level: Level 2 Teaching period(s): Semester 1 Offered by School of Mathematics Available as a free choice unit?: N

Prerequisite

Aims

The course unit unit aims to introduce basic ideas group theory 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.

Overview

This course unit provides an introduction to groups, one of the most important algebraic structures. It gives 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 MATH10101/MATH10111. Groups are a fundamental concept in mathematics, particularly in the study of symmetry and of number theory.

Learning outcomes

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

• Appreciate and use the basic definitions and properties of groups;
• Command a good understanding of the basic properties for a good range of examples;
• Understand and find simple proofs of results in group theory.

Assessment methods

• Other - 20%
• Written exam - 80%

Assessment Further Information

• Coursework; An in-class test, weighting within unit 20%
• 2 hours end of semester examination; Weighting within unit 80%

Syllabus

• Binary operations. Multiplication tables, associativity, commutativity, associative powers. [2 lectures]
• Groups. Definitions and examples (groups of numbers, the integers modulo n, symmetric groups, groups of matrices). [2]
• Subgroups. Subgroup criterion, cyclic subgroups, centralizer, centre, order of an element. [4]
• Cyclic groups. Subgroups of cyclic groups are cyclic, subgroups of finite cyclic groups. [1]
• Cosets and Lagrange's Theorem. [2]
• Homomorphisms and isomorphisms. Definition and examples, group theoretic properties. [2]
• Conjugacy. Conjugacy classes, conjugacy in symmetric groups, the class formula. [4]
• Normal subgroups. [2]
• Factor groups. [2]
• The First Isomorphism Theorem [1]

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

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

Study hours

• Lectures - 33 hours
• Independent study hours - 67 hours

Teaching staff

Peter Rowley - Unit coordinator