Automorphism groups of free metabelian nilpotent groups

Published in Algebra i Logika, Vol. 29, No. 6, pp. 746–751, November–December, 1990.     

Algebraic sets in metabelian groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = F_n is a free metabelian group of rank n; (2) G = W_n,k is a wreath product of free Abelian groups of ranks n and k. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007.     

Symmetric words in metabelian groups

An n-ary word w(x₁,…,x_n) is called n-symmetric for a group G if w(g₁,…,g_n) = w(g_{σ 1},…,g_{σ n}) for all g₁,…,g_n in G and all permu¬tations a in the symmetric group S_n. In this note we describe 2 and 3-symmetric words in free metabelian groups and metabelian groups of nilpotency class c, for arbitrary c.     

Dominions in quasivarieties of metabelian groups

The dominion of a subgroup H of a group A in a quasivariety ℳ is the set of all a ∈ A with equal images under all pairs of homomorphisms from A into every group in ℳ which coincide on H. The concept of dominion provides some closure operator on the lattice of subgroups of a given group. We study the closed subgroups with respect to this operator. We find a condition for the dominion of a divisible subgroup in quasivarieties of metabelian groups to coincide with the subgroup.     

Group rings with metabelian unit groups

Let F be a field of odd characteristic and G a group. In 1991 Shalev established necessary and sufficient conditions so that the unit group of the group ring FG is metabelian when G is finite. Here, in the modular case, we do the same without restrictions on G. In particular, new cases emerge when G contains elements of infinite order.     

D'Alembert's functional equations on metabelian groups

Summary. We extend d'Alembert's classical functional equation by replacing the domain of definition Bbb R {Bbb R} of the solutions by a metabelian group G and simultaneously replacing the group involution by an arbitrary involution of G. We find all complex valued solutions. In particular we show that the continuous solutions have the same form as in the abelian case if G is connected.     

Solvable absolute Galois groups are metabelian

We show that solvable absolute Galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion-free procyclic group. In particular, solvable absolute Galois groups are metabelian. Moreover, any field with solvable absolute Galois group G admits a non-trivial henselian valuation, unless each Sylow-subgroup of G is either procyclic or isomorphic to Z ₂⋊Z/2Z. A complete classification of solvable absolute Galois groups (up to isomorphism) is given. Oblatum 22-IV-1998 & 1-IX-2000?Published online: 30 October 2000     

The lattice of quasivarieties of metabelian groups

Let ℳ be a quasivariety of torsion-free groups satisfying the identity (∀x)(∀y)(x ²,y ²)=1. It is proved that the lattice of quasivarieties contained in ℳ has the power of the continuum. Supported by RFFR grant No. 93-011-1524. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 552–561, September–October, 1996.     

Universal equivalence of partially commutative metabelian groups

We state necessary and sufficient conditions for two partially commutative metabelian groups defined by trees to be universally equivalent.     

Subdirect products of finitely presented metabelian groups

There has been substantial investigation in recent years of subdirect products of limit groups and their finite presentability and homological finiteness properties. To contrast the results obtained for limit groups, Baumslag, Bridson, Holt and Miller investigated subdirect products (fibre products) of finitely presented metabelian groups. They showed that, in contrast to the case for limit groups, such subdirect products could have diverse behaviour with respect to finite presentability.We show that, in a sense that can be made precise, ‘most’ subdirect products of a finite set of finitely presented metabelian groups are again finitely presented. To be a little more precise, we assign to each subdirect product a point of an algebraic variety and show that, in most cases, those points which correspond to non-finitely presented subdirect products form a subvariety of smaller dimension.