Wolfgang Kimmerle

A Sylowlike theorem for integral group rings of finite solvable groups

By V(RG) we denote the units in RG, which have augmentation 1. The group of units in RG is then the product of the units in R and V(RG). A subgroup H of V(RG) with \H\ = |G| is called a group basis, provided the elements of H are linearly independent. This latter condition is automatic, provided no rational prime divisor of \H\ is a unit in R [1]. If H is a group basis, then RG = RH as augmented algebras and conversely. The object of this note is to prove the following

Finite groups of units and their composition factors in the integral group rings of the groups PSL(2,q)

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

Projective limits of group rings

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 ZGi. The basic topic of the paper is the question whether Γ may be a replacement of ZG. In particular, this is studied in connection with the isomorphism problem of integral group rings and with the conjecture of Zassenhaus that different group bases of ZG are conjugate within QG. Using such projective limits, a Cech style cohomology set yields obstructions for these conjectures to be true, if G is soluble. This is used to construct two non-isomorphic groups as projective limits such that the projective limits of the corresponding group rings are semi-locally isomorphic. On the other hand, it is shown that for special classes of groups certain p-versions of the Zassenhaus conjecture hold. These p-versions are weaker than the conjecture but still provide a strong positive answer to the Isomorphism problem. In particular, such p-versions hold when G has a nilpotent commutator subgroup or when G is a Frobenius or a 2-Frobenius group.

On the Normalizer Problem

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.

Autoequivalences of blocks and a conjecture of Zassenhaus

Abstract 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 automorphism of the integral group ring of PSL(2, p), p a rational prime, is given by a group automorphism followed by a conjugation in QPSL(2, p). In particular this proves a conjecture of Zassenhaus for these groups. Finally we show the same statement for a couple of other simple groups by different methods.

ON PRINCIPAL BLOCKS OF p-CONSTRAINED GROUPS

A theorem of K. W. Roggenkamp and L. L. Scott shows that for a finite group G with a normal p- subgroup containing its own centralizer, any two group bases of the integral group ring

On the cohomology of alternating and symmetric groups and decomposition of relation modules

\mathbb{Z} G

Irreducible components and isomorphisms of the Burnside ring

are conjugate in the units of

Mini-Workshop: Arithmetik von Gruppenringen

\mathbb{Z}_{p}G

On the Gruenberg–Kegel graph of integral group rings of finite groups

. Though the theorem presents itself in the work of others and appears to be needed, there is no published account. There seems to be a flaw in the proof, because a ‘theorem’ appearing in the survey [K. W. Roggenkamp, ‘The isomorphism problem for integral group rings of finite groups’, Progress in Mathematics 95 (Birkhauser, Basel, 1991), pp. 193--220], where the main ingredients of a proof are given, is false. In this paper, it is shown how to close this gap, at least if one is only interested in the conclusion mentioned above. Therefore, some consequences of the results of A. Weiss on permutation modules are stated. The basic steps of which any proof should consist are discussed in some detail. In doing so, a complete, yet short, proof of the theorem is given in the case that G has a normal Sylow p- subgroup. 2000 Mathematical Subject Classification: primary 16U60; secondary 20C05.