Abstract

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 N_𝓤(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, Out_c(G) = 1, have the normalizer property. Recently, Herman and Li have shown that Out_c(G) = 1 for a finite Blackburn group G. We show that Out_c(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.