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Normal automorphisms of a free pro-$$p$$-group in the variety $${\mathcal N}_2{\mathcal A}$$. (English. Russian original) Zbl 0976.20017
Algebra Logika 35, No. 3, 249-267 (1996); translation in Algebra Logic 35, No. 3, 139-148 (1996).
Summary: An automorphism of a (profinite) group is called normal if each (closed) normal subgroup is left invariant by it. An automorphism of an abstract group is $$p$$-normal if each normal subgroup of $$p$$-power index, where $$p$$ is a prime, is left invariant. Obviously, an inner automorphism of a group is normal and $$p$$-normal. For some groups, the converse was stated to be likewise true. N. Romanovskij and V. Boluts, for instance, established that for free solvable pro-$$p$$-groups of derived length 2, there exist normal automorphisms that are not inner. Let $${\mathcal N}_2$$ be the variety of nilpotent groups of class 2 and $$\mathcal A$$ the variety of Abelian groups. We prove the following results: (1) If $$p$$ is a prime number distinct from 2, then a normal automorphism of a free pro-$$p$$-group of rank $$\geq 2$$ in $${\mathcal N}_2{\mathcal A}$$ is inner (Theorem 1): (2) if $$p$$ is a prime number distinct from 2, then a $$p$$-normal automorphism of an abstract free $${\mathcal N}_2{\mathcal A}$$-group of rank $$\geq 2$$ is inner (Theorem 2).
##### MSC:
 2e+19 Limits, profinite groups 2e+37 Automorphisms of infinite groups 2e+11 Quasivarieties and varieties of groups 2e+29 Maximal subgroups
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