Group theory

Introduction

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This free OpenLearn course, Group theory, is an extract from the Open University course M303 Further pure mathematics [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] , a third level course that introduces important topics in the theory of pure mathematics including: number theory; the algebraic theory of rings and fields; and metric spaces. Students studying M303 develop their understanding of group theory and real analysis and see how some of the ideas are applied to cryptography and fractals.

Group theory consists of material from M303 Book B, Chapters 6 and 7 and has three sections in total. You should set aside about five hours to study each of the sections. The whole extract should take about 16 hours to study. The extract is a small part (around 4%) of a large course that is studied over eight months, and so can give only an approximate indication of the level and content of the full course.

The aim of this extract is to consolidate and build on the group theory presented at Open University level 2 of the curriculum in pure mathematics. It includes lots of problems and while studying you should always have pencil and paper at hand. It is relatively self-contained and should be reasonably easy to understand for someone with a sound knowledge of pure mathematics, such as could be gained from studying the Open University course M208 Pure mathematics. A few techniques and definitions are present in the extract without explanation.

Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. However, the extract isn't entirely representative of the module materials, because there are no explicit references to use of the M303 video material (although please note that the PDF may contain references to other parts of M303). In this extract, some illustrations have also been removed due to copyright restrictions.

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Section 1 describes how to construct a group called the direct product of two given groups, and then describes certain conditions under which a group can be regarded as the direct product of its subgroups.

Section 2 describes the key properties of the structure of cyclic groups, starting with a complete description of all cyclic groups.

Section 3 introduces the notion of a set of generators of a group and a set of relations among the generators. As an example, it looks at a family of finite groups called the dicyclic groups.