A conjectura de Zassenhaus

Abstract

In the mid 1960’s, the german mathematician Hans J. Zassenhaus, inspired by Gram Higman’s thesis and the work of Ian Hughes and Kenneth R. Pearson, stated several conjectures that changed radically the field of research in group rings. Since then many papers have been published about these conjectures, where either an affirmative answer is settled for some specific case, or some counterexamples are determined. Still, it is only very recently that Florian Eisele and Leo Margolis, based on some joint work with Ángel del Río, have found a counterexample for the only conjecture which has been left open. This fact motivated the writing of this dissertation. Our first goal is to revise the notions necessary to understand the conjectures, where they sit in the theory of group rings, and how they relate to representation theory. Then, we describe the counterexamples in an elementary way, in order to simplify the reading of some of their properties. For instance, we determine their character tables. We also point out, still omitting most of the details, which are the techniques involved in the proof that these groups are indeed counterexamples.