Normal automorphisms of free solvable pro-p-groups

An automorphism of a profinite group is called normal if it leaves invariant all (closed) normal subgroups. An automorphism of an abstract group is called p-normal if it leaves invariant each normal subgroup of p-power, where p is prime. An inner automorphism satisfies both of these conditions. Earlier, Romanovskii and Boluts [2] gave a description of normal automorphisms of a free solvable pro-p-group of derived length 2. That description implied, in particular, that the number of normal automorphisms in that group exceeds the number of inner ones. Here we prove that each normal automorphism of a free solvable pro-p-group of derived length ≥ 3 and a p-normal automorphism of an abstract free solvable group of derived length ≥ 2 are inner. © 1997 Plenum Publishing Corporation.