an:01334950
Zbl 0943.16012
Li, Yuanlin; Sehgal, S. K.; Parmenter, M. M.
On the normalizer property for integral group rings
EN
Commun. Algebra 27, No. 9, 4217-4223 (1999).
00058733
1999
j
16U60 20C05 16S34 20D30
finite groups; normalizer property; units; integral group rings; centers; isomorphism problem
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 \textit{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.
A.Zimmermann (Amiens)
Zbl 0141.02401