Ciclo de Palestras – UAMat/UFCG

**Open problems on units of integral group rings**

Angel Del Río (Universidad de Murcia – Espanha)

18 de maio de 2022 – 14h00 (horário de Brasília)

Videoconferência: https://meet.google.com/tew-ofqj-owr

Resumo: Let $R$ be a commutative ring and $G$ a finite group. The group ring $RG$ is the free $R$-module with basis $G$ with the product given by $(rg)(sh)=(rs)(gh)$ for $r,s\in R$ and $g,h\in G$. The group ring $RG$ encodes many properties of the group $G$ and the cases where $R$ is a field $K$ or the ring of integers $\Z$ is of particular interest because the finite dimensional $KG$-modules are basically the representations of $G$ over $K$ and the $\Z G$-modules encodes the integral representations of $G$. The group of units of $RG$ is denoted by $\U(RG)$ and the subgroup formed by units of augmentation 1 is denoted $V(RG)$.

The study of the structure of $\U(\Z G)$ was started by G. Higman in the 1940’s. He proved for example that if $G$ is abelian then all the torsion elements of $\U(\Z G)$ belong $\pm G$ and using this he proved that if $H$ is another group with $\Z G$ and $\Z H$ isomorphic as rings then $G$ and $H$ are isomorphic as groups. This was the start of the so called Isomorphism Problem for Group Rings which formally states the following:

$$RG\cong RH \Rightarrow G\cong H?$$

Higman result showed that the the group of units of $RG$ collects significant information about the ring $RG$ itself at least in the case where $R=\Z$.

Since the seminal work of Higman many researchers has study both the Isomorphism Problem and the structure of $\U(\Z G)$ yielding several nice techniques. The result of Higman and some other results suggested the so called Zassenhaus Conjectures which predicts that the finite subgroups of $V(\Z G)$ are conjugate in $\Q G$ of subgroups of $G$. This turns out to be false in general but true for many families of groups including all nilpotent finite groups. Other weaker versions of the Zassenhaus Conjecture try to describe the how the finite subgroups of $V(\Z G)$ are in comparison with the finite subgroups of $G$. This is the case for example of the Spectrum Problem which asks whether the set of orders of the torsion units of $V(\Z G)$ and $G$ coincide.

We will present a panorama of the known results and open problems on the Isomorphism Problem for group rings and the different approaches to the description of the finite subgroups of $V(\Z G)$.