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Class-preserving automorphisms and the normalizer property for Blackburn groups

2008; De Gruyter; Volume: 12; Issue: 1 Linguagem: Inglês

10.1515/jgt.2008.068

1435-4446

Martin Hertweck, Eric Jespers,

Rings, Modules, and Algebras

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.