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

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.

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

Zbl 0141.02401
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References:

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