Class-preserving Automorphisms and the Normalizer Property for Blackburn Groups

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 nontrivial. 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 a few 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.