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Normal automorphisms of free solvable pro-$$p$$-groups of derived length $$2$$. (English. Russian original) Zbl 0827.20039
Algebra Logic 32, No. 4, 239-243 (1993); translation from Algebra Logoka 32, No. 4, 441-449 (1993).
For an abstract group an automorphism is called normal if it leaves each normal subgroup invariant. A. Lubotsky [J. Algebra 63, No. 2, 494- 498 (1980; Zbl 0432.20025)] and 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 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.

##### MSC:
 20E18 Limits, profinite groups 20E36 Automorphisms of infinite groups 20F16 Solvable groups, supersolvable groups 20F28 Automorphism groups of groups 20E05 Free nonabelian groups