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BSc Mathematics with Financial Mathematics / Course details

Year of entry: 2021

Course unit details:Algebraic Structures 1

Unit code MATH20201 10 Level 2 Semester 1 Department of Mathematics No

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.

Pre/co-requisites

Unit title Unit code Requirement type Description
Foundations of Pure Mathematics A MATH10101 Pre-Requisite Compulsory
Foundations of Pure Mathematics B MATH10111 Pre-Requisite Compulsory

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.

Learning outcomes

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

- state the group axioms and identify frequently met examples of groups,
- define basic concepts in group theory, such as subgroups, conjugacy classes, cyclic groups, cosets, and factor groups,
- employ the subgroup criterion to determine whether certain subsets of a group are subgroups,
- describe fundamental properties of cosets and factor groups,
- identify the generators and subgroups of cyclic groups,
- determine conjugacy classes, cosets and factor groups in certain groups,
- state, prove and apply Lagrange's theorem.

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]

Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework; An in-class test, weighting within unit 20%
• End of semester examination; Weighting within unit 80%

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.

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

Study hours

Scheduled activity hours
Lectures 33
Independent study hours
Independent study 67

Teaching staff

Staff member Role
Marianne Johnson Unit coordinator