Normal relatively convex subgroups of solvable orderable groups

Orderable solvable groups in which every relatively convex subgroup is normal are studied. If such a class is subgroup closed than it is precisely the class of solvable orderable groups which are locally of finite (Mal’tsev) rank. A criterion for an orderable metabelian group to have every relatively convex subgroup normal is given. Examples of an orderable solvable group G of length three with periodic G/G′ and of an orderable solvable group of length four with only one proper normal relatively convex subgroup are constructed. To the memory of N. Ya. Medvedev Supported by RFBR (project No. 03-01-00320). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 291–308, May–June, 2009.     

Groups of formal power series are fully orderable

We construct an example of a fully orderable group that is not locally solvable. It is also shown that a free group is embedded in a fully orderable group. To meet these ends, use is made of a group of invertible formal power series with zero free term under composition. Supported by RFFR grant No. 96-01-00088. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 301–319, May–June, 1998.     

Normal automorphisms of a free pro-p-group in the varietyN 2 A

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, where p is prime, is left invariant. Obviously, the inner automorphism of a group will be normal and p-normal. For some groups, the converse was stated to be likewise true. N. Romanovskii 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 N₂ be the variety of nilpotent groups of class 2 and A the variety of Abelian groups. We prove the following results: (1) If p is a prime number distinct from 2, then the normal automorphism of a free pro-p-group of rank ≥2 in N₂A is inner (Theorem 1); (2) If p is a prime number distinct from 2, then the p-normal automorphism of an abstract free N₂A-group of rank ≥2 is inner (Theorem 2). Supported by RFFR grant No. 93-01-01508. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 249–267, May–June, 1996.     

Universally equivalent solvable groups

Every group that is finitely presented in the varietyA ⁿ of solvable groups. and is universally equivalent to a free group F_r(A ⁿ) in this variety, is embedded in the Cartesian degree of F₂(A ⁿ). All subgroups on a set of two generators in that Cartesian degree which are universally equivalent to F₂(A ⁿ) are determined. Free solvable and nilpotent groups are proved universally equivalent. Supported by RFFR grant No. 99-01-00567, and through the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 227–240, March–April, 2000.     

Asymptotic growth of averaged Dehn functions for nilpotent groups

It is proved that in any finite representation of any finitely generated nilpotent group of nilpotency class l ⩾ 1, the averaged Dehn function σ(n) is subasymptotic w.r.t. the function n^l+1. As a consequence it is stated that in every finite representation of a free nilpotent group of nilpotency class l of finite rank r ⩾ 2, the Dehn function σ(n) is Gromov subasymptotic. Supported by RFBR grant No. 04-01-00489. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 60–74, January–February, 2007.     

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.     

Conjugately dense subgroups of free products of groups with amalgamation

A subgroup having non-empty intersection with each class of conjugate elements of the group is said to be conjugately dense. It is shown that, under certain conditions, the number of conjugately dense subgroups in a free product with amalgamation is not less than some cardinal. As a consequence, P. Neumann’s conjecture in the Kourovka notebook (Question 6.38) is refuted. It is also stated that a modular group and a non-Abelian group of countable or finite rank possess continuum many pairwise non-conjugate conjugately dense subgroups. Supported by RFBR grant No. 03-01-00905. __________ Translated from Algebra i Logika, Vol. 45, No. 5, pp. 520–537, September–October, 2006.     

Irreducible Algebraic Sets in Metabelian Groups

We present the construction for a u-product G₁ ○ G₂ of two u-groups G₁ and G₂, and prove that G₁ ○ G₂ is also a u-group and that every u-group, which contains G₁ and G₂ as subgroups and is generated by these, is a homomorphic image of G₁ ○ G₂. It is stated that if G is a u-group then the coordinate group of an affine space Gⁿ is equal to G ○ F_n, where F_n is a free metabelian group of rank n. Irreducible algebraic sets in G are treated for the case where G is a free metabelian group or wreath product of two free Abelian groups of finite ranks. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 601–621, September–October, 2005. Supported by RFBR grant No. 05-01-00292, by FP “Universities of Russia” grant No. 04.01.053, and by RF Ministry of Education grant No. E00-1.0-12.     

Groups elementarily equivalent to a free 2-nilpotent group of finite rank

We give a characterization of groups elementarily equivalent to a free 2-nilpotent group of finite rank. Translated from Algebra i Logika, Vol. 48, No. 2, pp. 203–244, March–April, 2009.     

Abstract isomorphisms of quasiminimal groups

A connected solvable algebraic group is called minimal if its center is trivial and it has no proper connected subgroups with trivial center. We show that abstract isomorphisms of minimal groups defined over fields of characteristic zero are standard. Supported by RFFR grants Nos. 96-01-01678 and 96-01-01675, and by the State Committee for Higher Education of Russia, grant No. 2. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 567–586, September–October, 1996.     

Basis ranks of solvable quasivarieties of groups andl-groups

We point out quasivarieties of torsion-free solvable groups (and varieties of solvable l-groups) which are not generated by their all n-generated groups (l-groups), i.e., basis ranks for such are infinite. Supported by RFFR grant No. 99-01-00156. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 586–594, September–October, 2000.     

On one class of modules over integer group rings of locally solvable groups

We study a Z G-module A in the case where the group G is locally solvable and satisfies the condition min–naz and its cocentralizer in A is not an Artinian Z-module. We prove that the group G is solvable under the conditions indicated above. The structure of the group G is studied in detail in the case where this group is not a Chernikov group. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, No. 1, pp. 44–51, January, 2009.     

Free group of infinite unitriangular matrices

In the group of infinite unitriangular matrices over the field with two elements, a free subgroup of rank two is constructed which is a group of finite-automata transformations over a two-element alphabet. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 382–386, March, 2000.     

A Freiheitssatz for products of groups

Denote byB a class of solvable groups having a finite normal series with torsion-free Abelian factors, and by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaWefv3ySLgzgjxyRrxDYbqehuuDJXwAKbIrYf2A0vNC% aGGbaiqb-fa8czaaraaaaa!475E!\[\bar \mathfrak{B}\] a class of groups every finitely generated subgroup of which is approximated by {ie193-3}. We prove that if {ie193-4} is a free product with relations of groups A₁,…,A_n in the class {ie193-5}, where n>m and all relations are taken from the Cartesian subgroups, then there exist distinct indices i₁,…,i_n-m such that gp(A_i1,…,Ai_n-m)=A_i1 *…* Ai_n-m. A similar fact is established for solvable products with relations. Supported by RFFR grant No. 99-01-00567. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 354–367, May–June, 1999.     

Maximal quasivarieties of groups

We consider a lattice L_q(qG) of quasivarieties contained in the quasivariety qG, generated by a polycyclic-by-finite group G. It is proved that the lattice contains a finite set of coatoms (i.e., proper maximal elements) and that each of its elements distinct from qG is contained in some coatom. We construct an example of a finitely generated solvable group B of derived length 3, whose quasivariety lattice L_q(qB) is freed of coatoms. Supported by RFFR grant No. 96-01-00088, and by the RF Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 279–290, May–June, 1998.     

On identities of right alternative metabelian Grassmann algebras

The right alternative metabelian (solvable of index 2) Grassmann algebras of rank 1 and 2 are studied. A basis of identities of a right alternative metabelian Grassmann algebra of rank 1 is presented. Then it is proved that the variety generated by the indicated algebra has almost finite topological rank. It is also shown that the variety generated by a Grassmann algebra of rank 2 is not Spechtian. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 157–183, 2007.     

A sharper tits alternative for 3-manifold groups

The following theorem is proven. LetM be a closed, orientable, irreducible 3-manifold such that rankH ₁(M, ℤ/pℤ)≥3 for some primep. Then either π₁(M) is virtually solvable or it contains a free group of rank 2.     

Groups of finite non-Abelian sectional rank

We study non-Abelian locally finite groups and non-Abelian locally solvable groups of finite non-Abelian sectional rank and prove that their (special) rank is finite. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1324–1331, October, 1997.     

Nil-elements of index 2 in Mal’tsev algebras

It is proved that a Mal'tsev algebra over an associative commutative ring with 1, which contains 1/6 and is generated by a finite tuple of nil-elements of index 2, is nilpotent, and that an ideal of the Mal'tsev algebra over a field of characteristic 0, generated by nil-elements of index 2, is locally solvable. Supported by RFFR grant No. 96-01-01511. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 358–373, May–June, 1998.