On the normalizer problem. (English) Zbl 1063.16036

Summary: The normalizer problem of an integral group ring of an arbitrary group \(G\) is investigated. It is shown that any element of the normalizer \({\mathcal N}_{{\mathcal U}_1}(G)\) of \(G\) in the group of normalized units \({\mathcal U}_1(\mathbb{Z} G)\) 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 \({\mathcal N}_{{\mathcal U}_1}(G)\) equals \(G'\) and \({\mathcal N}_{{\mathcal U}_1}(G)/G\) is finitely generated if the torsion subgroup of the finite conjugacy group of \(G\) is finite.


16U60 Units, groups of units (associative rings and algebras)
16S34 Group rings
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
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