关于几乎可解群

本文推广了关于局部有限群的Asar定理及p.Hall—Kulatilaka,Kargapolov定理.     

Bounding essential arities of term operations in finitely decidable varieties

Let A be a finite algebra generating a finitely decidable variety and having nontrivial strongly solvable radical \({\tau}\). We provide an improved bound on the number of variables in which a term can be sensitive to changes within \({\tau}\). We utilize a multisorted algebraic construction, amalgamating the methods developed by Valeriote and McKenzie for the investigation of strongly abelian locally finite decidable varieties with those of Idziak for locally finite congruence modular finitely decidable varieties.     

Finite groups with a splitting automorphism of odd order

In this paper, we prove that a finite group with a splitting automorphism of odd order is solvable. By using this result, we prove that a locally finite group with a splitting automorphism of odd order is locally solvable.     

A strong Mal’cev condition for locally finite varieties omitting the unary type

We show that a finite algebra has a Taylor operation if and only if it has an operation satisfying a particular set of 6-ary Taylor identities. As a consequence we get the first strong Mal’cev condition for the family of locally finite varieties omitting the unary type. This is of interest to combinatorialists, as it is conjectured that a Constraint Satisfaction Problem defined by a core relational structure is polynomial time solvable exactly when a certain associated variety omits the unary type. Our result implies that the problem of deciding if a core relational structure meets this characterisation is itself in NP.     

Congruence extension,Hamiltonian and Abelian properties in locally finite varieties

We prove that in a locally finite variety with the congruence extension property, locally solvable congruences are central and locally solvable algebras are Hamiltonian. Also, we prove that a maximal subuniverse of a finite algebra in an Abelian variety is identical with an equivalence class of some congruence.Presented by H. P. Gumm.     

Locally graduated groups with complemented infinite nonprimary subgroups

The investigations of infinite groups with certain systems of complemented infinite subgroups, suggested by Chernikov, are continued. It is proved that an infinic locally graduated, nonprimary group with complemented infinite nonprimary subgroups is locally finite and solvable, and all of its nonprimary subgroups have complements if and only if it is not Chernikov.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 6, pp. 839–842, June, 1992.     

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.     

Generalization of fischer's theorem for distributive quasigroups

We generalize Fischer's well-known theorem which asserts that the right associated group of a finite distributive quasigroup is solvable. We prove that the associated group of a finitely generated distributive quasigroup is solvable and that the associated group of any distributive quasigroup is locally solvable.Translated from Matematicheskie Zametki, Vol. 23, No. 4, pp. 521–526, April, 1978.     

Factorization of Periodic Locally Solvable Groups by Locally Nilpotent and Nilpotent Subgroups

We establish a series of new results concerning periodic locally solvable and finite solvable groups G = AB with locally nilpotent or nilpotent subgroups A and B.     

Efficient solution of the word problem in slim varieties

Certain varieties similar to commutative semigroups are shown to have uniformly solvable word problem for all finite presentations by a confluence-completion method.Presented by H. P. Gumm.     

On Finite Basis Property for Joins of Varieties of Associative Rings

If all finitely generated rings in a variety of associative rings satisfy the ascending chain condition on two-sided ideals, the variety is called locally weak noetherian. If there is an upper bound on nilpotency indices of nilpotent rings in a variety, the variety is called a finite index variety. We prove that the join of a finitely based locally weak noetherian variety and a variety of finite index is also finitely based and locally weak noetherian. One consequence of this result is that if an associative ring variety is connected by a finite path in the lattice of all associative ring varieties to a finitely based locally weak noetherian variety then such variety is also finitely based and locally weak noetherian.     

Locally graded groups with complemented infinite nonabelian subgroups

Infinite nonabelian groups with complemented infinite nonabelian subgroups are investigated. It is proved that, under the condition of being locally graded, these groups are locally finite and solvable, and all nonabelian subgroups are complemented in them if and only if they are non-Chernikov groups.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, Nos. 7 and 8, pp, 1098–1100, July–August, 1991.     

A Generalization of Lie's Theorem

We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L.     

Varieties of rings,where all subdirectly irreducible finite rings are Armendariz ones

In this paper we study varieties of rings, where all subdirectly irreducible finite rings are Armendariz. We also describe the locally finite varieties of Armendariz rings.     

On internal formation theory

Summary This paper describes analogues to locally induced formations which are defined entirely within a given finite solvable group. The methods are used to construct complements to normal subgroups and to determine when all such complements are conjugate.Work on this paper was supported by a grant from the National Science Foundation.     

Examples of locally finite groups

In this paper we construct examples of nonextremal locally finite groups for which every finite unsolvable subgroup is not embedded in any proper infinite subgroup. We also give an example of a locally finite simple group, any proper infinite subgroup of which is bigrade solvable.Translated from Matematicheskie Zametki, Vol. 13, No. 1, pp. 103–106, January, 1973.The author wishes to express his deep gratitude to Yu. M. Gorchakov for help and valuable advice in the preparation of the paper.     

On the existence of f-local subgroups in a group

We prove the existence of infinite subgroups with nontrivial locally finite radicals and of locally finite subgroups in the groups with almost finite almost solvable elements of prime orders and in the groups with generally finite elements.     

Locally almost solvable groups of finite non-Abelian rank

It is proved that a nonperiodic, locally almost solvable group of finite non-Abelian 0-rank has finite (special) rank.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 477–482, April, 1990.