Description of concrete subgroups of the unit group of an integral group ring (FWOTM783)

On the edge of ring theory and group theory, there is a way to "stitch" a group and a ring together, called a group ring. This gives us a ring-theoretical way of dealing with groups and can be used to study the representations of a group. A special (and interesting) case arises if we look at the group ring of the integers and a finite group. The study of the so called unit group - the elements by which we can "divide" in the group ring - is crucial to the isomorphism problem. The question here is if the structure of the group ring determines the structure of the group, or in other words, when two group rings of groups over the same ring are isomorphic, are the groups themselves isomorphic or not? In this project we will tackle some problems that arise when we study this unit group. Some of these problems include finding the structure of the subgroup generated by a special kind of these units, seeing whether they describe a large part of the unit group and finding elements without relations between them to construct so called free (semi)groups. These free (semi)groups can be used to study the growth of a group, a way to describe how big a group is without looking at its cardinality. These integral group rings are also examples of so called Z-orders; these are rings that resemble some aspects of the integers and are of major importance in other mathematical domains such as number theory. We will also consider the problems outlined in this more general context.

Keywords:Mathematics, ring theory, group theory

Disciplines:Applied mathematics in specific fields not elsewhere classified