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

Requisites

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.

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%

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.

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]

Recommended reading

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

Feedback methods

Tutorials will provide a place for student worked examples to be marked and discussed providing feedback on performance and understanding.

Study hours

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

Teaching staff

Ralph Stohr - Unit coordinator


Data source is Central CUIP

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