MA2014: Algebra I: Groups

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA2014
External Subject Code 100405
Number of Credits 10
Level L5
Language of Delivery English
Module Leader Professor Roger Behrend
Semester Autumn Semester
Academic Year 2022/3

Outline Description of Module

A group is a mathematical entity with a simple structure: a set with a binary operation satisfying certain axioms. Groups are ubiquitous throughout mathematics, helping us describe symmetries in nature. Some examples of groups are integers under the operation of addition, permutations under the operation of composition, and invertible square matrices under the operation of matrix multiplication. Some real-world examples in which groups appear are symmetries of crystals, wallpaper patterns, Islamic geometric patterns, error detecting codes such as ISBN numbers on books, and the geometry of the spacetime we inhabit.

Some of the basic definitions and concepts in group theory were introduced in the Year One module MA1005 Foundations of Mathematics I. This Year Two module will build upon these concepts in order to provide an in-depth study of group theory and its most fundamental results.

On completion of the module a student should be able to

  • State and apply the group axioms.
  • Understand the concepts of abelian groups, subgroups, generators, homomorphisms, isomorphisms, kernels, images, conjugacy, cosets, normal subgroups and quotient groups.
  • Prove fundamental group theorems, including Lagrange’s theorem, the first isomorphism theorem and Cayley’s theorem.
  • Work with certain examples of groups, such as cyclic, dihedral and symmetric groups.

How the module will be delivered

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

  • Weekly face to face classes (e.g. labs, lectures, exercise classes)
  • Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills that will be practised and developed

Skills:
Appreciation of the general prevalence and significance of groups in mathematics. Ability to understand and apply the basic concepts of group theory.

Transferable Skills:
Recognition of the power of mathematical abstraction and the importance of mathematical rigour. Critical thinking and problem-solving skills. Ability to present work in a scholarly manner.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 100 Algebra I: Groups 2

Syllabus content

  1. Groups, abelian groups, subgroups, generators, homomorphisms, isomorphisms, kernels, images, conjugacy, cosets, normal subgroups and quotient groups.
  2. Fundamental group theorems, including Lagrange’s theorem, the first isomorphism theorem and Cayley’s theorem.
  3. Examples of groups, including cyclic, dihedral and symmetric groups.
  4. Concepts related to the symmetric group, including cycles, transpositions and signs.

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