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.


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
Full Text: DOI


[1] DOI: 10.1080/00927879308824756 · Zbl 0788.16024
[2] DOI: 10.1016/0021-8693(66)90018-4 · Zbl 0141.02401
[3] Coleman D.B., Proc. Amer. Math. Soc 5 pp 511– (1964)
[4] DOI: 10.1016/0022-4049(87)90028-4 · Zbl 0624.20024
[5] Hertweck M., A solution of the isomorphism problem (1997)
[6] Mazur M., Expo. Math 13 pp 433– (1995)
[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
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