ABELIANIZATION OF THE UNIT GROUP OF AN INTEGRAL GROUP RING

For a finite group G and U := 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/ U' of U. We pose questions on the connections between the exponent of G I G' and the exponent of U/U' as well as between the ranks of the torsion-free parts of Z(U), the center of U, and U/U'. We show that the units originating from known generic constructions of units in ZG are well-behaved under the projection from U to U/ U' 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 (Bachle et al. 2018b).