Wolfgang Kimmerle

• Professor
• University of Stuttgart

Let G be a finite group and denote by \({\mathbb {Z}}G\) its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided \({\mathbb {Z}}G \cong {\mathbb {Z}}H\) and G is q-constrained. If addit...

Textbook for Bachelor students of the 3rd semester in engineering sciences, physics and related fields

In this paper we study the behavior of the first Zassenhaus conjecture (ZC1) under direct products, as well as the General Bovdi Problem (Gen-BP), which turns out to be a slightly weaker variant of (ZC1). Among other things, we prove that (Gen-BP) holds for Sylow tower groups, and so in particular for the class of supersolvable groups. (ZC1) is est...

We consider the question whether a Sylow like theorem is valid in the normalized units of integral group rings of finite groups. After a short survey on the known results we show that this is the case for integral group rings of Frobenius groups. This completes work of M.A. Dokuchaev, S.O. Juriaans and V. Bovdi and M. Hertweck. We analyze projectiv...

We describe the main questions connected to torsion subgroups in the unit group of integral group rings of finite groups and algorithmic methods to attack these questions. We then prove the Zassenhaus Conjecture for Amitsur groups and prove that any normalized torsion subgroup in the unit group of an integral group of a Frobenius complement is isom...

We describe the main questions connected to torsion subgroups in the unit group of integral group rings of finite groups and algorithmic methods to attack these questions. We then prove the Zassenhaus Conjecture for Amitsur groups and prove that any normalized torsion subgroup in the unit group of an integral group of a Frobenius complement is isom...

The prime graph question asks whether the Gruenberg-Kegel graph of an integral group ring $\mathbb Z G$ , i.e. the prime graph of the normalised unit group of $\mathbb Z G$ coincides with that one of the group $G$. In this note we prove for finite groups $G$ a reduction of the prime graph question to almost simple groups. We apply this reduction to...

We consider the question whether a Sylow like theorem is valid in the normalized units of integral group rings of finite groups. After a short survey on the known results we show that this is the case for integral group rings of Frobenius groups. This completes work of M.A. Dokuchaev, S.O. Juriaans and V. Bovdi and M. Hertweck. We analyze projectiv...

The main part of this article is a survey on torsion subgroups of the unit group of an integral group ring. It contains the major parts of my talk given at the conference “Groups, Group Rings and Related Topics” at UAEU in Al Ain October 2013. In the second part special emphasis is layed on p - subgroups and on the open question whether there is a...

Every four years, leading researchers gather to survey the latest developments in all aspects of group theory. Since 1981, the proceedings of those meetings have provided a regular snapshot of the state of the art in group theory and helped to shape the direction of research in the field. This volume contains selected papers from the 2013 meeting h...

Im folgenden Beitrag wird versucht darzulegen, dass Primzahlen in der Mathematikausbildung, sowohl in der Schule als auch an der Universität, eine herausragende Rolle spielen können und sollen.

The article provides a survey on the unit group of integral group rings. It leads from classical theorems to recent advances on this topic. The focus is on large subgroups, i.e. subgroups of finite index in U, as well as on units of finite order. A short look at the methods used in the proofs is given. The interplay between group theory, representa...

The object of this article are torsion subgroups of the normalized unit group V(ZG) of the integral group ring ZG of a finite group G. For specific subgroups W we study the Gruenberg–Kegel graph Π(W). It is shown that the central elements of an isolated subgroup U of a group basis H of ZG are the normalized units of its centralizer ring CZG(U). Mor...

Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is isomorphic to a subgroup of $G$. Furthermore, it is shown that a composition factor of a finite subgroup of $V(\ma...

Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is isomorphic to a subgroup of $G$. Furthermore, it is shown that a composition factor of a finite subgroup of $V(\mat...

In this paper, we prove that each component of the Burnside ring of a finite group is the soluble component of the Burnside ring of a Weyl subgroup of its corresponding group. We show that groups with isomorphic Burnside rings have the same sublattice of soluble normal subgroups and the same spectrum. This gives for the alternating groups, the spor...

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 thesize of the group allowed excellent discussions amongst all participants. V...

It was conjectured by H. Zassenhaus that a torsion unit of an integral group ring of a finite group is conjugate to a group element within the rational group algebra. The object of this note is the computational aspect of a method developed by I.S. Luthar and I.B.S. Passi which sometimes permits an answer to this conjecture. We illustrate the metho...

Abstract. The combinatorial surfaces with doubly transitive automorphism groups are classified. This is established by classifying the automorphism groups of combinatorial surfaces which act doubly transitively on the vertices of the surface. The doubly transitive automorphism groups of combinatorial surfaces are the symmetric group S 4 , the alte...

An automorphism of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who's important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer of G in the units of the integr...

Let G be a finite non-Abelian simple group. In the first part we consider the question whether ℂG determines G up to isomorphism. This question is closely related to a recent conjecture of B. Huppert that G is determined up to a direct Abelian factor by its set of ordinary character degrees. We sketch a proof that a finite simple group is determine...

The object of this article is to examine a conjecture of Zassenhaus and certain variations of it for integral group rings of sporadic groups. We prove the Q-variation and the Sylow variation for all sporadic groups and their automorphism groups. The Zassenhaus conjecture is established for eighteen of the sporadic simple groups, and for all automor...

The aim of this article is to lead the reader on a journey through the representation theory of finite groups of Lie type and Hecke algebras. We will present some basic results obtained in recent years, explain the ideas behind them, and give lots of examples; proofs are usually omitted but we provide explicit references to an extensive bibliograph...

The object of this note is the structure of the normalizer of a group basis of the group ring RG of a finite group G, where R = Z or more generally in the situation when R is G-adapted. This means that R is an integral domain of characteristic zero in which no prime divisor of |G| is invertible.

The object of this note is the structure of the normalizer of a group basis of the group ring RG of a finite group G, where R = ℤ or more generally in the situation when R is G-adapted. This means that R is an integral domain of characteristic zero in which no prime divisor of |G| is invertible.

The Atlas of Finite Groups by J. H. Conway, R. A. Parker, S. P. Norton and the editors of this book, was published in 1985, and has proved itself to be an indispensable tool to all researchers in group theory and many related areas. The present book is the proceedings of a conference organised to mark the 10th anniversary of the publication of the...

In this paper, we show that for every finite group with cyclic Sylow p-subgroups the principal p-block B is rigid with respect to the trivial simple module. This means that each autoequivalence which fixes the trivial simple module fixes the isomorphism class of each finitely generated B-module.As a consequence each augmentation preserving automorp...

A finite group G may be written as a projective limit of certain quotients Gi. Denote by Γ the corresponding projective limit of the integral group rings Gi. The basic topic of the paper is the question whether Γ may be a replacement of G. In particular, this is studied in connection with the isomorphism problem of integral group rings and with the...

The object of this article is to show that a Jordan-Hölder class structure of a finite group determines abelian Hall subgroups of the group up to isomorphism. The proof uses this classification of the finite simple groups.

The object of this section is the proof of further properties of the class sum correspondence. In particular we study the so-called powermap, i.e the behaviour of corresponding class sums under powers, and collect properties of a finite group determined by its character table. The consequences with respect to the isomorphism problem are the content...

Throughout this section R is an integral domain of characteristic zero, G a finite group and no prime divisor of |G| is invertible in R . K denotes a field containing R. Therefore by the previous sections we have between group bases in RG a dass sum correspondence.

The object of this section is to establish further classes of groups which satisfy the Zassenhaus Conjecture Variation 2, cf. section X.

Let G = An+1, Ap+2,Sp, or Sp+1, where p ≥ 5 is a prime. It is shown that if M is an irreducible G-module of characteristic q≠p, then dim , where d = |NG(P):P|, P a Sylow p-subgroup of G. This is used to show that relation cores for these groups always decompose. This answers a question of Gruenberg and Roggenkamp, who had observed that the relation...

The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a f...