On the Normalizer Problem

Abstract

In this paper the normalizer problem of an integral group ring of an arbitrary group G is investigated. It is shown that any element of the normalizer 1 G of G in the group of normalized units 1 G is determined by a finite normal subgroup. This reduction to finite normal subgroups implies that the normalizer property holds for many classes of (infinite) groups, such as groups without non-trivial 2-torsion, torsion groups with a normal Sylow 2-subgroup, and locally nilpotent groups. Further it is shown that the commutator of 1 G equals G and 1 G /G is finitely generated if the torsion subgroup of the finite conjugacy group of G is finite.  2002 Elsevier Science

Class-preserving automorphisms and the normalizer property for Blackburn groups

“…denotes the nth term of the upper central series of U. The question whether in general Z y ðUÞ c N U ðGÞ was finally answered in the a‰rmative in [8]. As a consequence of the present work we obtain that Z y ðUÞ c ZðUÞG.…”

“…For any u A N U ðGÞ, there exists a finite normal subgroup N of G such that ug À1 A RN for all g A suppðuÞ, by [8,Theorem 1.4]. Hence, it is enough to consider an element u in N U ðGÞ which is contained in RN for some finite normal subgroup N of G.…”

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

“…Since u ∈ N G , Theorem 1 of Jespers et al (2002) says that u = gw where g ∈ G and w ∈ T . If T is Abelian, w ∈ C T and we are done, so assume that T is a Q-group.…”

Central units in salternative loop rings

“…Under certain conditions, any unit of ZL can be factored as the product of an element of and an element of the loop ring whose support is in the torsion subloop of L [2, §XII.1]. As with group rings ( [4], [3]), it turns out that any central unit can always be so factored. Theorem 2.1.…”