Small ranks of central unit groups of integral group rings of alternating groups

We prove that the ranks of central unit groups of integral group rings of alternating groups of degrees greater than 38 are at least 11. The presented tables contain the ranks of all central unit groups of integral group rings of alternating groups of degrees at most 200. In particular, for every r ∈ {0, …, 10}, we obtain the complete list of integers n such that the central unit group of the integral group ring of the alternating group of degree n has rank r.     

Free products of abelian groups in the unit group of integral group rings

We classify finite groups which are such that the unit group of the integral group ring has a subgroup of finite index which is a non-trivial free product of abelian groups.

     

Central units of integral group rings of nilpotent groups

In this paper a finite set of generators is given for a subgroup of finite index in the group of central units of the integral group ring of a finitely generated nilpotent group.

     

Torsion units in integral group rings of solvable groups

A weaker version of the Zassenhaus conjecture for torsion units in integral group rings ZG is proved if G is either abelian-by-polycyclic or metabelian. As a consequence we obtain Bovdi's conjecture for torsion units in ZG for metabelian groups     

Torsion units in integral group rings of metacyclic groups

It is proved that if G is a split extension of a cyclic p-group by a cyclic p′-group with faithful action then any torsion unit of augmentation one of $\text{Z}$G is rationally conjugate to a group element. It is also proved that if G is a split extension of an abelian group A by an abelian group X with (|A|, |X|) = 1 then any torsion unit of $\text{Z}$G of augmentation one and order relatively prime to |A| is rationally conjugate to an element of X.     

Generators of large subgroups of the unit group of integral group rings

This work is supported in part by NSERC Grant OGP0036631, Canada, and CNPq, Brasil     

On isomorphisms between centers of integral group rings of finite groups

For finite nilpotent groups and , and a -adapted ring (the rational integers, for example), it is shown that any isomorphism between the centers of the group rings and is monomial, i.e., maps class sums in to class sums in up to multiplication with roots of unity. As a consequence, and have identical character tables if and only if the centers of their integral group rings and are isomorphic. In the course of the proof, a new proof of the class sum correspondence is given.

     

On the normalizer problem for integral group rings of torsion groups

In this paper, we investigate the normalizer property for the integral group ring of a torsion group. We show that this property holds for locally finite nilpotent groups. A necessary and sufficient condition for this property to hold for any torsion group is also given. This research was supported in part by a research grant from the Natural Sciences and Engineering Research Council of Canada.     

On the isomorphism problem for integral group rings of finite groups

Partly supported by the Deutsche Forschungsgemeinschaft and the National Science Foundation.     

Primitive central idempotents of finite group rings of symmetric groups

Let be a prime. We denote by the symmetric group of degree , by the alternating group of degree and by the field with elements. An important concept of modular representation theory of a finite group is the notion of a block. The blocks are in one-to-one correspondence with block idempotents, which are the primitive central idempotents of the group ring , where is a prime power. Here, we describe a new method to compute the primitive central idempotents of for arbitrary prime powers and arbitrary finite groups . For the group rings of the symmetric group, we show how to derive the primitive central idempotents of from the idempotents of . Improving the theorem of Osima for symmetric groups we exhibit a new subalgebra of which contains the primitive central idempotents. The described results are most efficient for . In an appendix we display all primitive central idempotents of and for which we computed by this method.

     

On the generators of subgroups of unit groups of group rings

In this paper we find the generators of a subgroup of finite index in the unit group of the integral group ring of the metacyclic group of orderpq given byG=(a,x:a ^p=1=x ^q ,xax ⁻¹=a ^f ), wherep is an odd prime,q>2 a divisor ofp-1, andf belongs to the exponentq modulop.