Class-preserving automorphisms and the normalizer property for Blackburn groups

Abstract

For a group $G$, let $U$ be the group of units of the integral group ring $\mathbb{Z}G$. The group $G$ is said to have the normalizer property if $\text{N}_U(G)=\text{Z}(U)G$. 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, $\text{Out}_c(G)=1$, have the normalizer property. Recently, Herman and Li have shown that $\text{Out}_c(G)=1$ for a finite Blackburn group $G$. We show that $\text{out}_c(G)=1$ 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 of hypercentral units of $U$ is contained in $\text{Z}(U)G$.