# Oberwolfach Reports

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**Volume 4, Issue 4, 2007, pp. 3209–3240**

**DOI: 10.4171/OWR/2007/55**

Published online: 2008-09-30

Mini-Workshop: Arithmetik von Gruppenringen

Eric Jespers^{[1]}, Wolfgang Kimmerle

^{[2]}, Zbigniew Marciniak

^{[3]}and Gabriele Nebe

^{[4]}(1) Vrije Universiteit Brussel, Bruxelles, Belgium

(2) Universität Stuttgart, Germany

(3) Warsaw University, Poland

(4) RWTH Aachen, Germany

The mini workshop ``Arithmetic of group rings'' was attended by 16 participants from Belgium, Brazil, Canada, Germany, Hungary, Israel, Italy, Romania and Spain. The expertise was a good mixture between senior and young researchers. It was a very stimulating experience and the size of the group allowed excellent discussions amongst all participants. Very fruitful were the problem sessions, resulting in the problems listed at the end of this report.

\bigskip \noindent The main highlights of the conference were: \begin{itemize} \item The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio. \item Hertweck's solution of the first Zassenhaus conjecture for finite metacyclic groups. \item the description of special subgroups of the unit group of integral group rings, such as the hypercentre and the finite conjugacy centre, and the relation with respect to the normalizer of the trivial units. \item discussion of the present state of art via several survey talks presented and the problem sessions. \end{itemize}

\bigskip \noindent The group $G$ determines its integral group ring $\Z G$ and its group $V(\Z G)$ of normalized units. Several talks addressed the interplay of the cohomological properties of these three objects. Further topics included twisted group rings, group rings over local rings, polynomial growth and identities, orders and semigroup rings, Lie structure, representation-theoretic and algorithmic methods.

\bigskip \noindent The most important open problems suggested in the conference are: \begin{itemize} \item Unit groups of integral group rings $\Z G$. \begin{enumerate} \item Construction of subgroups of finite index (see also Problem 14 and 18) \item The construction of specific subgroups by units of a given type (Problems 7 and 28). \item Specific properties of the unit group especially when $G$ is infinite (see Problems 6, 15, 18, 27, 29, 30). \end{enumerate} \item Torsion part of the unit group \begin{enumerate} \item The description of torsion units and torsion subgroups, in particular with respect to integral group rings of finite non-soluble groups and of infinite groups (see Problems 4, 9, 10, 13, 19, 20). \item The first Zassenhaus conjecture (see also Problems 21, 22) . \item The modular isomorphism problem, i.e. the question whether a finite $p$ - group is determined by $\mathbb{F}_p G$ up to isomorphism (see Problems 8, 31). \end{enumerate} \end{itemize}

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Jespers Eric, Kimmerle Wolfgang, Marciniak Zbigniew, Nebe Gabriele: Mini-Workshop: Arithmetik von Gruppenringen. *Oberwolfach Rep.* 4 (2007), 3209-3240. doi: 10.4171/OWR/2007/55