On the Zassenhaus Conjecture and the subgroups of finite index of the group of units of integral group rings. (FWOTM839)

Given a finite group G={g1,g2,...,gn}, the integral group ring ZG of G over the integers Z, is defined as the ring of all linear combinations of the form z1g1+z2g2+...+zngn, where the coefficients z1,z2,...,zn are integers. The addition is defined componentwise and the multiplication is the extension of the group multiplication. Let U(ZG) denote the group of units of ZG, i.e. the group of invertible elements of ZG. This group of units has been a theme of interest for a long time, however to obtain a complete description of it in terms of generators and relations seems to be a difficult task. One fundamental and still open question in this context is the so-called Zassenhaus Conjecture. It states that every unit of finite order of U(ZG) is conjugate to an element of G by a unit in the rational group algebra QG. Although this conjecture is easy to explain, it is extremely difficult. The main objective of the project is to prove or disprove the Zassenhaus Conjecture. With this investigation I would like to develop new techniques which might contribute to the knowledge of the Zassenhaus Conjecture. Another relevant question on the study of integral group rings consists in obtaining a complete description of the group of units U(ZG), for a finite non-abelian group G, in terms of generators and relations. Few progress have been obtain in this way, we shall investigate the subgroups of finite index in U(ZG).

Keywords:Zassenhaus Conjecture, Mathematics

Disciplines:General mathematics