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**Mini-workshop: Arithmetic of group rings. Abstracts from the mini-workshop held November 25 – December 1, 2007.
(Mini-workshop: Arithmetik von Gruppenringen.)**
*(English)*
Zbl 1177.16002

Introduction: 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.

The main highlights of the conference were: – The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio; – Hertweck’s solution of the first Zassenhaus conjecture for finite metacyclic groups; – 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; – discussion of the present state of art via several survey talks presented and the problem sessions.

The group \(G\) determines its integral group ring \(\mathbb{Z} G\) and its group \(V(\mathbb{Z} G)\) 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.

The most important open problems suggested in the conference are: Unit groups of integral group rings \(\mathbb{Z} G\): Construction of subgroups of finite index (see also Problem 14 and 18). The construction of specific subgroups by units of a given type (Problems 7 and 28). Specific properties of the unit group especially when \(G\) is infinite (see Problems 6, 15, 18, 27, 29, 30). Torsion part of the unit group: 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). The first Zassenhaus conjecture (see also Problems 21, 22). 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).

The main highlights of the conference were: – The complete calculation of the projective Schur subgroup of the Brauer group by Aljadeff and del Rio; – Hertweck’s solution of the first Zassenhaus conjecture for finite metacyclic groups; – 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; – discussion of the present state of art via several survey talks presented and the problem sessions.

The group \(G\) determines its integral group ring \(\mathbb{Z} G\) and its group \(V(\mathbb{Z} G)\) 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.

The most important open problems suggested in the conference are: Unit groups of integral group rings \(\mathbb{Z} G\): Construction of subgroups of finite index (see also Problem 14 and 18). The construction of specific subgroups by units of a given type (Problems 7 and 28). Specific properties of the unit group especially when \(G\) is infinite (see Problems 6, 15, 18, 27, 29, 30). Torsion part of the unit group: 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). The first Zassenhaus conjecture (see also Problems 21, 22). 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).

### MSC:

16-06 | Proceedings, conferences, collections, etc. pertaining to associative rings and algebras |

20-06 | Proceedings, conferences, collections, etc. pertaining to group theory |

16S34 | Group rings |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |

16U60 | Units, groups of units (associative rings and algebras) |

00B05 | Collections of abstracts of lectures |