Locally nilpotent groups with the minimal condition on centralizers

It is proved that locally nilpotent groups with the minimal condition on centralizers are hypercentral, and that the Fitting subgroup of a group with the minimal condition on centralizers of normal subgroups is nilpotent. Supported by RFFR grant No. 96-01-00358. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 270–278, May–June, 1998.     

Lattices with unique irreducible decompositions

We define the concept of a minimal decomposition in a lattice, and prove that all the currently known lattices with unique irreducible decompositions are in fact lattices with minimal ones. Also, the characterization of a class of lattices with minimal decompositions is given. A new proof of the Crawley-Dilworth characterization theorem for the class of coalgebraic strongly coatomic lattices with unique irreducible decompositions obtains as a consequence. In memory of Victor A. Gorbunov Supported by the RF State Committee of Higher Education (1998 project), by RFFR grant No. 99-01-00485, and jointly by RFFR and DFG grants Nos. 96-01-00097 and 436113/2670 respectively. Translated fromAlgebra i Logika, Vol. 39, No. 1, pp. 93–103, January–February, 2000.     

The intersection of Sylow subgroups in finite groups

In Theorem 1, letting p be a prime, we prove: (1) If G=S_n is a symmetric group of degree n, then G contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2, 2), (2, 4), (2, 8)}, and (2) If H=A_n is an alternating group of degree n, then H contains two Sylow p-subgroups with trivial intersection iff (p, n) ∉ {(3, 3), (2, 4)}. In Theorem 2, we argue that if G is a finite simple non-Abelian group and p is a prime, then G contains a pair of Sylow p-subgroups with trivial intersection. Also we present the corollary which says that if P is a Sylow subgroup of a finite simple non-Abelian group G, then ‖G‖>‖P‖². Supported by RFFR grants Nos. 93-01-01529, 93-01-01501, and 96-01-01893, and by International Science Foundation and Government of Russia grant RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 4, pp. 424–432, July–August, 1996.     

Minimal permutation representations of finite simple exceptional groups of typesG 2 andF 4

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G₂ and F₄. Supported by RFFR grant No. 96-01-01893, the program “Universities of Russia,” and by International Science Foundation and Government of Russia grant No. RPC300. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 663–684, November–December, 1996.     

Minimal permutation representations of finite simple exceptional twisted groups

A minimal permutation representation of a group is its faithful permutation representation of least degree. Here the minimal permutation representations of finite simple exceptional twisted groups are studied: their degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, are found. We can thus assert that, modulo the classification of finite simple groups, the aforesaid parameters are known for all finite simple groups. Supported by RFFR grant No. 96-01-01893, through the program “Universities of Russia”, and by grant No. RPC300 of ISF and the Government of Russia. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 17–35, January–February, 1998.     

Solvable elimination of ramification in extensions of discretely valued fields

An analog of the Abhyankar lemma is proved for an arbitrary finite separable extensions of discretely valued fields with a functional residue field. Supported by RFFR grants Nos. 96-01-01675 and 96-01-01678, and by RF Goskomvuz grant No. 2. Translated fromAlgebra i Logika, Vol. 37, No. 1, pp. 63–87., January–February, 1998.     

The second initial boundary value problem for the gravitation-gyroscopic wave equation in exterior domains

We study the existence and uniqueness of the solution to the second initial boundary value problem for the gravitation-gyroscopic wave equation in an exterior multiply connected domain with various types of conditions at infinity. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 40–57, July, 1996. This research was partially supported by the Russian Foundation for Basic Research under grant No. 96-01-01411.     

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.     

Classification up to cobordism of manifolds with simple action of ℤ/p

We deal with quasi-complex manifolds with an action of the group ℤ/p such that the set of fixed points of this action has a trivial normal bundle. The set of cobordism classes of these manifolds is described in terms of the coefficients of the formal group of geometric cobordisms and in terms of characteristic numbers. We also establish the relationship between this work and relevant papers containing a solution of this problem in some particular cases. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 260–268, February, 1998. The author is greatly indebted to his advisor Professor V. M. Bukhshtaber for his active interest in this work and useful discussions. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01404.     

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.     

Conformal type and isoperimetric dimension of Riemannian manifolds

The notion of conformal isoperimetric dimension is introduced. For Riemannian manifolds, connections between its conformal isoperimetric dimension and its conformal type are established. Translated fromMatematicheskie Zametki, Vol. 63, No. 3, pp. 379–385, March, 1998. The authors are greatly indebted to N. A. Zorich for her help in preparing the computer version of this paper. This research was partially supported by the Russian Foundation for Basic Research under grants No. 96-01-01218 and No. 96-01-00901.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.     

Groups with finitely many linear orders

It is proved that there exist orderable groups having exactly 6 and 14 distinct linear orders. For any natural number k, we construct examples of orderable groups on which 2(4k+3) linear orders are defined. Supported by RFFR grant No. 96-01-00088. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 176–200, March–April, 1999.     

The polynomial growth of colength of varieties of Lie algebras

It is proved that the colength of every API-variety of Lie algebras grows polynomially, and we give a number of examples in which the colength grows more rapidly than any polynomial function does. These indicate that for many of the important varieties of Lie algebras, such as varieties of solvable algebras of derived length 3, varieties generated by some infinite-dimensional simple algebras of Cartan type, or by certain Katz-Mudi algebras, the growth of colength will be superpolynomial. Supported by RFFR grants No. 96-01-00146 and No. 96-15-96050. Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 161–175, March–April, 1999.     

A property of the character table for a finite group

A character table X of a finite group is broken up into four squares: A, B, C, and D. We establish relations via which ranks of the matrices inX are connected. In particular, ifX is an l × l-matrix, A is an s × t-matrix, and, moreover, the squares A and C are opposite, thenr(A)=r(C) + s + t − l; here.r(M) is the rank of a matrix M. Associated with such each block ofX is some integral nonnegative parameter m, and we have m=0 iff A, B, C, and D are active fragments ofX. Supported by RFFR grant No. 96-01-00488. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 273–279, May–June, 2000.     

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.     

Element orders in coverings of symmetric and alternating groups

We prove that if the factor group H=G/N of a finite group G is isomorphic to a symmetric or alternating group of degree m, where m≥5 and N≠1, then G has an element whose order is distinct from any element’s order in H. Supported by RFFR grant No. 96-01-01893. Translated fromAlgebra i Logika, Vol. 38, No. 3, pp. 296–315, May–June, 1999.     

Superintuitionistic logics and the projective beth property

It is proved that in superintuitionistic logics, the projective Beth property follows from the Craig interpolation property, but the converse does not hold. A criterion is found which allows us to reduce the problem asking whether the projective Beth property is valid in superintuitionistic logics to suitable properties of varieties of pseudoboolean algebras. It is shown that the principle of variable separation follows from the projective Beth property. On the other hand, the interpolation property in a logic L implies the projective Beth property in Δ(L). Supported by RFFR grants No. 96-01-01552 and No. 99-01-00600. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 680–696, November–December, 1999.     

Stably Definable Classes of Theories

A question is studied as to which properties (classes) of elementary theories can be defined via generalized stability. We present a topological account of such classes. It is stated that some well-known classes of theories, such as strongly minimal, o-minimal, simple, etc., are stably definable, whereas, for instance, countably categorical, almost strongly minimal, ω-stable ones, are not. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 583–600, September–October, 2005. Supported by RFBR grant Nos. 02-01-00540 and 05-01-00411, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.