Abelianization of the unit group of an integral group ring

Abstract

For a finite group $G$ and $U:=U(\mathbb{Z}G)$, the group of units of the integral group ring of $G$, we study the implications of the structure of $G$ on the abelianization $U/U^{\prime }$ of $U$. We pose questions on the connections between the exponent of $G/G^{\prime }$ and the exponent of $U/U^{\prime }$ as well as between the ranks of the torsion-free parts of $Z(U)$, the center of $U$, and $U/U^{\prime }$. We show that the units originating from known generic constructions of units in $\mathbb{Z}G$ are well-behaved under the projection from $U$ to $U/U^{\prime }$ 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].

16 pages

References in corpus (2)

• Integral group rings of solvable groups with trivial central units
• Integral group rings with all central units trivial: solvable groups

Cited by in corpus (1)

• On integral decomposition of unipotent elements in integral group rings