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 ≤ 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.
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Gérard Endimioni, Automorphisms Fixing Every Normal Subgroup of a Nilpotent-by-Abelian Group. Rend. Sem. Mat. Univ. Padova 120 (2008), pp. 73–77DOI 10.4171/RSMUP/120-5