On the normalizer property for integral group rings. (English) Zbl 0943.16012

Li, Yuanlin; Sehgal, S. K.; Parmenter, M. M.

Commun. Algebra 27, No. 9, 4217-4223 (1999).

The authors discuss the following property: A finite group \(G\) has the normalizer property if a unit \(u\) in the integral group ring \(\mathbb{Z} G\) normalizes \(G\) if and only if there is an element \(g\in G\) so that \(u\cdot g\) is in the centre of \(\mathbb{Z} G\). This question is of importance for the isomorphism problem for integral group rings.
The authors prove that if the intersection of all non-normal subgroups of \(G\) is not reduced to the identity, then \(G\) has the normalizer property. The class of these groups was discussed by N. Blackburn [J. Algebra 3, 30-37 (1966; Zbl 0141.02401)] and they are of rather restricted structure. The authors use essentially the above result of Blackburn. M. Hertweck gave examples for groups which do not have the normalizer property.

Reviewer: A.Zimmermann (Amiens)

MSC:

16U60	Units, groups of units (associative rings and algebras)
20C05	Group rings of finite groups and their modules (group-theoretic aspects)
16S34	Group rings
20D30	Series and lattices of subgroups

Keywords:

finite groups; normalizer property; units; integral group rings; centers; isomorphism problem

Citations:

Zbl 0141.02401

References:

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[2]	DOI: 10.1016/0021-8693(66)90018-4 · Zbl 0141.02401
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[7]	Sehgal S.K., Topics in Group Rings (1978) · Zbl 0411.16004
[8]	Sehgal S.K., Units in Integral Group Rings (1993) · Zbl 0803.16022