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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:
20E18 Limits, profinite groups
20E36 Automorphisms of infinite groups
20E10 Quasivarieties and varieties of groups
20E28 Maximal subgroups
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