Abelianization of the unit group of an integral group ring

Abstract

For a finite group G and U : = U ( Z G ) , 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 / 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 Z G 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 (Bächle et al. 2018b).