On the Normalizer Problem

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...

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Bibliographic Details
Published in:Journal of algebra 2002-01, Vol.247 (1), p.24-36
Main Authors: Jespers, E., Juriaans, S.O., de Miranda, J.M., Rogerio, J.R.
Format: Article
Language:English
Subjects:
automorphism
group ring
normalizer
unit
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Summary: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.
ISSN:0021-8693
1090-266X
DOI:10.1006/jabr.2001.8724