Automorphisms Fixing Every Normal Subgroup of a Nilpotent-by-Abelian Group

Abstract

Among other things, we prove that the group of automorphisms fixing every normal subgroup of a (nilpotent of class $c$)-by-abelian group is (nilpotent of class $\le c$)-by-metabelian. In particular, the group of automorphisms fixing every normal subgroup of a metabelian group is soluble of derived length at most $3$. An example shows that this bound cannot be improved.