Normal automorphisms of free solvable pro-\(p\)-groups of derived length \(2\) (Q1346909)

For an abstract group an automorphism is called normal if it leaves each normal subgroup invariant. \textit{A. Lubotsky} [J. Algebra 63, No. 2, 494- 498 (1980; Zbl 0432.20025)] and \textit{A. Lue} [ibid. 64, No. 1, 52-53 (1980; Zbl 0435.20015)] have proved that in a free group a normal automorphism is inner and \textit{V. A. Roman'kov} [Sib. Mat. Zh. 24, No. 4, 138-149 (1983; Zbl 0518.20028)] has proved that also in the free solvable groups of derived length \(\geq 2\) each normal automorphism is inner. In the present paper the authors consider the case of normal automorphisms of solvable pro-\(p\)-groups \(F\) of derived length 2 and finite rank. Here a topological automorphism of a profinite group is called normal if it leaves each (closed) normal subgroup invariant. They give a description of the group \(\text{Aut}_NF\) of normal automorphisms of \(F\) and prove that there are normal automorphisms which are not inner. In fact they prove that the group \(\text{Aut}_NF/\text{Inn }F\) is not finitely generated. They also get as a corollary of the proof that if the automorphisms \(\varphi\) of \(F\) is such that, if for each \(f\in F\) there exists \(g\in F\) with \(\langle f\varphi\rangle=g^{-1}\langle f\rangle g\) then \(\varphi\) is inner. For the proofs the authors use the Magnus representation of the free metabelian group as a group of \(2\times 2\) matrices adapted conveniently here.