Partially commutative metabelian groups: centralizers and elementary equivalence

For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Supported by RFBR (project No. 09-01-00099). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 309–341, May–June, 2009.     

Structure of coordinate groups for algebraic sets in partially commutative nilpotent groups

The results obtained deal in algebraic geometry over partially commutative class two nilpotent ℚ-groups, where ℚ is a field of rationals. It is proved that two arbitrary non-Abelian partially commutative class two nilpotent ℚ-groups are geometrically equivalent. A necessary and sufficient condition of being universally geometrically equivalent is specified for two partially commutative class two nilpotent ℚ-groups. Algebraic sets for systems of equations in one variable, as well as for some special systems in several variables, are described. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Translated from Algebra i Logika, Vol. 48, No. 2, pp. 378–399, May–June, 2009.     

The arithmetical hierarchy of nilpotent torsion-free groups

Previously, N. Khisamiev proved that all {ie172-1} Abelian torsion-free groups are {ie172-2}. We prove that for the class of nilpotent torsion-free groups, the situation is different: even the quotient group F of a {ie172-3} nilpotent group of class 2 by its periodic part may fail to have a {ie172-4}. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 308–313, May–June, 1996.     

Nilpotent groups admitting an almost regular automorphism of order four

We consider locally nilpotent periodic groups admitting an almost regular automorphism of order 4. The following are results are proved: (1) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then (a) the subgroup {ie176-1} contains a subgroup of m-bounded index in {ie176-2} which is nilpotent of m-bounded class, and (b) the group G contains a subgroup V of m-bounded index such that the subgroup {ie176-3} is nilpotent of m-bounded class (Theorem 1); (2) If a locally nilpotent periodic group G admits an automorphism ϕ of order 4 having exactly m<∞ fixed points, then it contains a subgroup V of m-bounded index such that, for some m-bounded number f(m), the subgroup {ie176-4}, generated by all f(m) th powers of elements in {ie176-5} is nilpotent of class ≤3 (Theorem 2). Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000. Translated fromAlgebra i Logika, Vol. 35, No. 3, pp. 314–333, May–June, 1996.     

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.     

Hochschild cohomologies for associative conformal algebras

We introduce the concept of Hochschild cohomologies for associative conformal algebras. It is shown that the second cohomology group of a conformal Weyl algebra with values in any bimodule is trivial. As a consequence, we derive that the conformal Weyl algebra is segregated in any extension with nilpotent kernel. Supported by RFBR grant No. 05-01-00230 and via SB RAS Integration project No. 1.9. __________ Translated from Algebra i Logika, Vol. 46, No. 6, pp. 688–706, November–December, 2007.     

Large nilpotent subgroups of finite simple groups

Orders and the structure of large nilpotent subgroups in all finite simple groups are determined. In particular, it is proved that if G is a finite simple non-Abelian group, and N is some of its nilpotent subgroups, then |N|²<|G|. Supported through FP “Integration” project No. 274, by RFFR grant No. 99-01-00550, by International Soros Education Program for Exact Sciences (ISEP) grant No. S99-56, and by a SO RAN grant for Young Scientists, Presidium Decree No. 83 of 03/10/2000. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 526–546, September—October, 2000.     

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.     

Infinite groups with Abelian centralizers of involutions

The article contains two characterizations of projective linear groups PGL₂(P) over a locally finite field P of characteristic 2: the first is defined in terms of permutation groups, and the second, in terms of a structure of involution centralizers. One of the two is used to prove the existence of infinite groups which are recognizable by the set of their element orders. In memory of Viktor A. Gorbunov Supported by RFFR grant No. 99-01-00550. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 74–86, January–February, 2000.     

Rational sets in finitely generated nilpotent groups

We deal with a class of rational subsets of a group, that is, the least class of its subsets which contains all finite subsets and is closed under taking union. a product of two sets, and under generating of a submonoid by a set. It is proved that the class of rational subsets of a finitely generated nilpotent group G is a Boolean algebra iff G is Abelian-by-finite. We also study the question asking under which conditions the set of solutions for equations in groups will be rational. It is shown that the set of solutions for an arbitrary equation in one variable in a finitely generated nilpotent group of class 2 is rational. And we give an example of an equation in one variable in a free nilpotent group of nilpotency class 3 and rank 2 whose set of solutions is not rational. Supported by RFFR grant No. 98-01-00932. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 379–394, July–August, 2000.     

The join of varieties with nilpotent intersection

It is proved that the join of finitely based varieties with nilpotent intersection has a finite basis of identities. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 228–242, March–April, 1996.     

Identities for Lie superalgebras with a nilpotent commutator subalgebra

It is proved that the exponential growth rate of identities in any superalgebra with a nilpotent commutator subalgebra over a field of zero characteristic is an integer. Supported by RFBR (project Nos. 07-01-00080 and 06-01-00485) and by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-1983.2008.1). Translated from Algebra i Logika, Vol. 47, No. 5, pp. 617–645, September–October, 2008.     

Centers of Mal’tsev and alternative algebras

Let ϕ be an associative commutative ring with unity containing 1/6. Let A and B be a free Mal’tsev and a free alternative ϕ-algebras on a set of k≥6 free generators, respectively. We construct nonzero homogeneous elements of degree 7 belonging to an annihilatorAnnA of A, and nonzero homogeneous elements of degree 7 belonging to the center Z(B) of B. It is shown that a nilpotent Mal’tsev algebra of index 8 on a set of 6 generators has no faithful representation. Supported by RFFR grant No. 96-01-01511, and by the Program “Universities of Russia: Fundamental Research.” Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 613–635, September–October, 1999.     

Finite groups admitting an automorphism of prime order with central fixed points

In 1957, Higman showed that a Lie algebra admitting a fixed-point-free automorphism is nilpotent, and that an analogous result also holds for a finite soluble group. Two years later, Thompson proved that a finite group having a fixed-point-free automorphism of prime order is soluble, and consequently nilpotent. Generalizing that situation, a few years ago, Kharchenko set up a conjecture on the solubility of a Lie algebra L admitting an automorphism of prime order whose fixed points lie in the center of L. A similar conjecture applies also with finite groups. Here we affirm the latter for the case where the order p of an automorphism is equal to 2 and deny it for all p>3. Supported by RFFR grants Nos. 93-01-01501 and 96-01-01893. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 699–708, November–December, 1996.     

Groups containing an element that permutes with a finite number of its conjugates

We study a group G containing an element g such that C_G(g)∩g^G is finite. The nonoriented graph Γ is defined as follows. The vertex set of Γ is the conjugacy class g^G. Vertices x and y of the graph G are bridged by an edge iff x≠y and xy=yx. Let Γ₀ be some connected component of G. On a condition that any two vertices of Γ₀ generate a nilpotent group, it is proved that a subgroup generated by the vertex set of Γ₀ is locally nilpotent. Supported by the RF State Committee of Higher Education. Translated fromAlgebra i Logika, Vol. 37, No. 6, pp. 637–650, November–December, 1998.     

Metric relations in groups of tree automorphisms

For groups of tree automorphisms, we couch an analog for a dimension that generalizes the ordinary dimension to the case of groups of direct power. On a Grigorchuk 2-group, that generalized dimension has the additivity property. Generalized dimensions of centralizers of elements in the Grigorchuk group are computed. The dual notion of a codimension is introduced. The relationship between the codimension of a subgroup and the dimension of this subgroup's centralizer is explicated. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 338–357, May–June, 1998.     

On Hardy’s uncertainty principle for connected nilpotent Lie groups

In (Kaniuth and Kumar in Math. Proc. Camb. Phil. Soc. 131, 487–494, 2001) Hardy’s uncertainty principle for was generalized to connected and simply connected nilpotent Lie groups. In this paper, we extend it further to connected nilpotent Lie groups with non-compact centre. Concerning the converse, we show that Hardy’s theorem fails for a connected nilpotent Lie group G which admits a square integrable irreducible representation and that this condition is necessary if the simply connected covering group of G satisfies the flat orbit condition.     

Lie rings admitting an automorphism of order 4 with few fixed points

Lie rings that admit an automorphism of order 4 with few fixed points are considered. For a Lie ring (algebra) L admitting an automorphism of order 4 with a finite number m of fixed points (with a finite-dimensional subalgebra of fized points of dimension m), it is proved that the subring 4L (algebra L) contains an ideal M with a subring of m-bounded index in the additive group of M (a subalgebra of m-bounded codimension), which is nilpotent of class bounded by some constant. It is also shown that, under the same premise, the factor-ring 4L/M (factor-algebra L/M) contains a subring of m-bounded index in the additive group of 4L/M (a subalgebra of m-bounded codimension), which is nilpotent of class ≤2. Moreover, L has a subring of m-bounded index in the additive group of L (a subalgebra of m-bounded codimension), which is soluble of derived length bounded by a constant. Supported by RFFR grant No. 94-01-00048 and by ISF grant NQ7000. Translated fromAlgebra i Logika, Vol. 35, No. 1, pp. 41–78, January–February, 1996.     

Groups with elementary Abelian centralizers of involutions

An involution i of a group G is said to be almost perfect in G if any two involutions of i^G the order of a product of which is infinite are conjugated via a suitable involution in i^G. We generalize a known result by Brauer, Suzuki, and Wall concerning the structure of finite groups with elementary Abelian centralizers of involutions to groups with almost perfect involutions. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 75–82, January–February, 2007.     

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.