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

Abstract

Let Z n ( ) denote the n'th term of the upper central series of the unit group of G and Z Z Z( )= ∪ n=1 Z n ( ). It is shown that if the set of torsion elements of G forms a subgroup T and Z Z Z ) C (T), then T is either an Abelian 2-group or a Q-group.Moreover, Z Z Z( )⊆ G · C (T) whenever G is an FC group.

On hypercentral units in integral group rings

“…The next two examples show that the central height of the unit group U can be arbitrarily high (as was already noted in [31], [32]). …”

“…We formulate it in a slightly stronger form as follows (cf. also [31], [32]). Recall from Example 7.1 that for an arbitrary Q*-group G, the hypercenter Z y ðUðZGÞÞ still has the expected description.…”

“…[1], [2], [14], [29]- [32]. Before describing the present work, we briefly comment on this development.…”

“…Li and Parmenter [31], [32] started to investigate the hypercentral units in ZG where G is not necessarily torsion, concentrating on what happens if Z 2 ðUÞ does not centralize all torsion elements of G.…”

Hypercentral Unit Groups and the Hyperbolicity of a Modular Group Algebra

“…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. In [6, Chapter VI] Hertweck extended these results to group rings RG of periodic groups G over G−adapted rings R. From its exposition is clear that the containement of Z ∞ (Γ) in the normalizer N Γ (G) is an important property.…”