Normal automorphisms of a free pro-\(p\)-group in the variety \({\mathcal N}_2{\mathcal A}\). (English. Russian original) Zbl 0976.20017

Gupta, Ch.; Romanovskij, N. S.

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

Keywords:

normal automorphisms; inner automorphisms; free solvable pro-\(p\)-groups; varieties of nilpotent groups; \(p\)-normal automorphisms