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Class-preserving automorphisms and the normalizer property for Blackburn groups
1Institut für Geometrie und Topologie, Fachbereich Mathematik, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany. E-mail: firstname.lastname@example.org
2Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium. E-mail: email@example.com
Citation Information: Journal of Group Theory. Volume 12, Issue 1, Pages 157–169, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: 10.1515/JGT.2008.068, November 2008
- Published Online:
For a group G, let 𝓤 be the group of units of the integral group ring ℤG. The group G is said to have the normalizer property if N𝓤(G) = Z(𝓤)G. It is shown that Blackburn groups have the normalizer property. These are the groups which have non-normal finite subgroups, with the intersection of all of them being non-trivial. Groups G for which class-preserving automorphisms are inner automorphisms, Outc(G) = 1, have the normalizer property. Recently, Herman and Li have shown that Outc(G) = 1 for a finite Blackburn group G. We show that Outc(G) = 1 for the members G of certain classes of metabelian groups, from which the Herman–Li result follows.
Together with recent work of Hertweck, Iwaki, Jespers and Juriaans, our main result implies that, for an arbitrary group G, the group Z∞(𝓤) of hypercentral units of 𝓤 is contained in Z(𝓤)G.