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 NU1(G) of G in the group of normalized units U1(ZG) 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 NU1(G) equals G′ and NU1(G)/G is finitely generated if the torsion subgroup of the finite conjugacy group of G is finite.