Some Results on Hypercentral Units in Integral Group Rings

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.

The Upper Central Series of the Unit Group of an Integral Group Ring

“…We note that even if G is an FC group, the above equality may fail. For example, if G = a x a 2 n = 1 x −1 ax = a −1 n ≥ 3 , it is not hard to check Z = Z 2 by using Theorem 2.7 of Li and Parmenter (2003). In this paper we extend all results about Z 2 in Li and Parmenter (2003) to Z .…”

“…However, the situation when G is not torsion is far less clear. In Proposition 2.5 of Li and Parmenter (2003), the inequality Z ⊆ G · Z was shown to continue to hold for some classes of groups while Theorem 2.3 of Li and Parmenter (2003) presented an additional class of groups which satisfied the weaker inequality Z ⊆ G · C T (where T is the set of torsion elements of G). The main result (Theorem 3.5) of Li and Parmenter (2003) showed that if T is a subgroup of G and Z 2 C T , then T is either an Abelian group of exponent 4 or a Q-group.…”

“…It was pointed out in Li and Parmenter (2003) that the equality Z = Z 2 does not hold in general when G is an arbitrary group. We note that even if G is an FC group, the above equality may fail.…”

Hypercentral Unit Groups and the Hyperbolicity of a Modular Group Algebra

“…Arora and Passi in [2] then proved that Z ∞ (Γ) = Z(Γ)T , where T denotes the torsion subgroup of Z ∞ (Γ). These results were extended to torsion groups by Li [10] and Li,Parmenter [11]. In [12], [13] they presented some contributions to the problem for non-periodic groups.…”

Hypercentral units in alternative loop rings

“…Specifically, we establish the following theorem. We also refer the reader to [5] where some of the results of this paper are established for group rings.…”