ALGB Seminar: Leo Margolis

15/02/2018 - 11:00

The Zassenhaus Conjecture II - A Counterexample

Abstract:
Hans J. Zassenhaus conjectured in 1974 that any unit of finite order in the integral group ring of a finite group G is conjugate in the rational group algebra to an element of the form ±g for some g ∈ G. Though proven e.g. for nilpotent or cyclic-by-abelian groups the conjecture does not hold in general and in a series of two talks I am going to present the first counterexamples recently found in collaboration with F. Eisele.

The existence of the counterexample is equivalent to showing the existence of a certain module M over an integral group ring ZG. Since classifying all modules of ZG is a hopeless task we first use general methods to approximate M via a rational module and then via p-adic modules. It then remains to consider the genus class group of ZG to obtain the existence of M. These general arguments allow to boil down the question to character and group theoretic questions which can eventually be solved by elementary calculations.