MA0213: Groups

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA0213
External Subject Code G100
Number of Credits 10
Level L5
Language of Delivery English
Module Leader Professor Roger Behrend
Semester Autumn Semester
Academic Year 2015/6

Outline Description of Module

A group consists of a set together with a binary operation which satisfies certain axioms. Many important classes of mathematical objects can be regarded as groups, some examples being symmetry transformations together with the operation of composition, integers together with the operation of addition, and invertible real matrices together with the operation of matrix multiplication.

This module will provide an introduction to some of the fundamental concepts of group theory. In particular, various general definitions and theorems will be studied, and then illustrated using specific examples.

On completion of the module a student should be able to

  • State the group axioms.
  • Understand the concepts of abelian groups, subgroups, homomorphisms, isomorphisms, cosets, normal subgroups and quotient groups.
  • Prove basic group theorems, such as 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

27 - 50 minute lectures.

Some handouts will be provided in hard copy and on Learning Central, but students will be expected to take notes of lectures.

Students will also be expected to undertake at least 50 hours of private study, including preparation of solutions to given exercises.

Skills that will be practised and developed

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.

How the module will be assessed

Formative assessment will be carried out by means of regular homework exercises. Feedback to students on their progress towards learning outcomes will be provided.

Summative assessment will be the written examination at the end of the module. This will give students the opportunity to demonstrate their overall achievement of learning outcomes. It will also enable them to provide evidence of the higher levels of knowledge and understanding required for above-average marks. The examination paper will have a choice of three from four equally-weighted questions.

Assessment Breakdown

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

Syllabus content

  • Groups, abelian groups, subgroups, homomorphisms, isomorphisms, conjugacy, cosets, normal subgroups and quotient groups.
  • Basic group theorems, including Lagrange’s theorem, the first isomorphism theorem and Cayley’s theorem.
  • Examples of groups, including cyclic, dihedral and symmetric groups.
  • Concepts related to the symmetric group, including cycles, transpositions and signs.

Essential Reading and Resource List

See Background Reading List

Background Reading and Resource List

Dummit, D. S., and Foote, R. M., Abstract Algebra (Third Edition), Wiley, 2004.

Ledermann, W.,  and Weir,  A.,  Introduction to Group Theory (Second Edition), Longman, 1996.


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