Abelianization of the unit group of an integral group ring

Abstract

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/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 [BJJ + 18].The following variations of (2) appear natural for V : (R1) Is rank V /V ′ = rank Z(V )? (R2) Assume Z(V ) is finite. Is V /V ′ also finite?The above questions address the free subgroup of V /V ′ , while the ones listed next try to get hold of its torsion elements. Denote by exp H, the exponent of a group H, i.e., the greatest common divisor of the orders of torsion elements of H.

The lower central series of the unit group of an integral group ring

“…A partial study of the above problem for a finite group G, namely, the study of the quotient γ 1 (V(ZG))/γ 2 (V(ZG)), i.e, V(ZG)/V(ZG) ′ has been taken up in [BMM20].…”