Free course

Group theory

More about this course

Course description

Course content

Course reviews

This free course consolidates and builds on group theory studied at OU level 2 or equivalent. 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. Section 3 introduces the notion of a set of generators of a group and a set of relations among the generators. It then looks at a family of finite groups, the dicyclic groups.

Course learning outcomes

After studying this course, you should be able to:

  • apply the Internal Direct Product Theorem in simple cases
  • decide whether a given group is cyclic, and given a finite cyclic group, find a generator for a subgroup of a given order
  • express a given finite cyclic group as the direct product of cyclic groups of prime power order and, given two direct products of cyclic groups, determine whether or not they are isomorphic
  • express products of elements of a group defined by generators and relations in appropriate standard form
  • recognise the dihedral and dicyclic groups when described using a standard form.

You can start this course right now without signing-up. Click on any of the course content sections below to start at any point in this course.
If you want to be able to track your progress, earn a free Statement of Participation, and access all course quizzes and activities, sign-up.

Course content

Expand all course content

For further information, take a look at our frequently asked questions which may give you the support you need.

Have a question?

Take your learning further

Making the decision to study can be a big step, which is why you'll want a trusted University. The Open University has nearly 50 years’ experience delivering flexible learning and 170,000 students are studying with us right now. Take a look at all Open University courses.

If you are new to university level study, find out more about the types of qualifications we offer, including our entry level Access courses and Certificates.

Not ready for University study then browse over 900 free courses on OpenLearn and sign up to our newsletter to hear about new free courses as they are released.

Every year, thousands of students decide to study with The Open University. With over 120 qualifications, we’ve got the right course for you.

Request an Open University prospectus

M303_2