Automorphisms fixing every normal subgroup of a nilpotent-by-abelian group

TL;DR

This paper investigates the structure of automorphisms that fix all normal subgroups in certain groups, proving they are nilpotent-by-metabelian and establishing bounds on their complexity.

Contribution

It characterizes the automorphism groups fixing all normal subgroups in nilpotent-by-abelian and metabelian groups, providing new structural insights and bounds.

Findings

Automorphisms fixing all normal subgroups form a nilpotent-by-metabelian group.

In metabelian groups, such automorphisms are soluble with derived length at most 3.

An example demonstrates the bound on derived length cannot be improved.

Abstract

Among other things, we prove that the group of automorphisms fixing every normal subgroup of a nilpotent-by-abelian group is nilpotent-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.