Learn about group theory and the structure of cyclic groups in this free course. Discover how to apply the Internal Direct Product Theorem and determine if a group is cyclic. Find out how to express a group as a direct product and recognize dihedral and dicyclic groups.
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.