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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.

MSC:

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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