Abstract: Initiated by G. Higman in 1940 research on the unit group of the integral group ring ZG of a finite group G has uncovered many interes- ting interactions between ring, group, representation and number theory. A conjecture of H. Zassenhaus from 1974 stated that any unit of finite order in ZG should be as trivial as one can possibly expect. More precisely it should be conjugate in the rational group algebra QG to an element of the form ±g for some g ∈ G.
I will recall some history of the problem and related questions and then present a recently found counterexample. The existence of the counterexam- ple is equivalent to showing the existence of a certain module over an integral group ring. Considering intermediate problems by variation of the coefficient ring allows to boil down the conjecture for a certain class of groups to ques- tions which can be solved by elementary calculations.
This is joint work with Florian Eisele.
