Abelianization Of The Unit Group Of An Integral Group Ring

Abstract

For a finite group G and U VD U.ZG/, the group of units of the integral group ring of G, we study the implications of the structure of G on the abelianization U=U0 of U. We pose questions on the connections between the exponent of G=G0 and the exponent of U=U0 as well as between the ranks of the torsion-free parts of Z.U/, the center of U, and U=U0. We show that the units originating from known generic constructions of units in ZG are well-behaved under the projection from U to U=U0 and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from (Bächle et al. 2018b).