Finite subgroups and quotients of the unit group of the integral group ring (FWOTM819)

Using both group and ring theory, we can make a special ring (the group ring) using a group, such that the multiplicative structure of the ring resembles that of the group. This gives us a natural way of studying the group itself, since there are many helpful techniques in ring theory. We are very much interested in the so called unit group (i.e. the elements by which we can "divide"). This group is very important in several major contemporary problems, such as (but not exclusively) the normalizer problem, the isomorphism problem... The last one is the simple question whether if two group rings are isomorphic, are the groups that determine these rings isomorphic? Though simple to pose, it has proven hard to find a counterexample. Only as recent as 1998 did Hertweck discover a counterexample, although the question was posed in 1940. In this project we wish to discover more about the precise structure of the unit group and in particular how the lattice of subgroups behaves. In particular we will pay attention to some special subgroups (generated by generic units) and look "how close" these subgroups are to being the entire unit group and (finite) quotients of this group. Lastly, we will investigate whether (under some assumptions) the unit group is a (semi-direct) product of a torsion-free normal subgroup with the original group. This would give a much better picture of the unit group and also it would give a positive answer to the isomorphism problem for such groups.

Keywords:integral group ring, subgroups, quotient, unit group, mathematics

Disciplines:General mathematics