No abstract

ABSTRACT. In this note we investigate the hypercentral units in integral group rings ZG, where G is not necessarily torsion. One of the main results obtained is the following (Theorem 3.5): if the set of torsion elements of G is a subgroup T of G and if Z 2 (U) is not contained in C U (T ), then T is either an Abelian group of exponent 4 or a Q * group. This extends our earlier result on torsion group rings.

No abstract

Abstract. In this paper we give new constructions of central units that generate a subgroup of finite index in the central units of the integral group ring ZG of a finite group. This is done for a very large class of finite groups G, including the abelian-by-supersolvable groups.Let G be a finite group. When G is abelian it is well known (and due to Bass and Milnor; see [1] and [9, Theorem 12.7 and Theorem 13.1]) that the Bass cyclic units of the integral group ring ZG generate a subgroup of finite index in Z(U(ZG)), the group of (central) units of ZG. In [4], it was shown that if G is a nilpotent finite group of nilpotency class n, then the group b (n) is of finite index in Z(U(ZG)).Here b (n) is defined recursively as follows. We denote by Z i the i-th centre of G. For any x ∈ G and Bass cyclic unit b ∈ Z x , put b (1) = b, and, for 2 ≤ i ≤ n, putRecently, Ferraz and Simón [2] gave a construction of central units in case G is meta-(cyclic of prime order) that generate a subgroup of finite index. They actually provided an independent set of generators.In this note we continue these investigations and provide new constructions of central units that generate a subgroup of finite index in the central units of ZG and this for a very large class of finite groups G, including the abelian-by-supersolvable groups. The proof and construction relies on a beautiful paper of Olivieri, del Río and Simón [6] in which they give an explicit construction of the primitive central idempotents of rational group algebras QG of the so-called strongly monomial groups (for example, abelian-by-supersolvable groups). Our proof thus uses a completely different method from the one used in [4] for finite nilpotent groups. In Section 1 we will recall the necessary background on this. In Section 2 we prove the main results for strongly monomial groups. In Section 3 we prove a general result that yields generators for central units from subgroups and quotient groups. An application is given for solvable Frobenius groups.

No abstract