Characterization of groups with generalized chernikov periodic part

A characterization of groups with generalized Chernikov periodic part is obtained first in the class of groups without elements of order 2 and then without this restriction. Two author’s theorems for periodic groups are generalized to mixed groups. Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 270–275, February, 2000.     

The property of being equationally Noetherian for some soluble groups

Let B be a class of groups A which are soluble, equationally Noetherian, and have a central series A = A₁ ⩾ A₂ ⩾ … A_n ⩾ … such that ⋂A_n = 1 and all factors A_n/A_n+1 are torsion-free groups; D is a direct product of finitely many cyclic groups of infinite or prime orders. We prove that the wreath product D ≀ A is an equationally Noetherian group. As a consequence we show that free soluble groups of arbitrary derived lengths and ranks are equationally Noetherian. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 46–59, January–February, 2007.     

On the Artin exponent of some rational groups

A finite group G is called rational if all its irreducible complex characters are rational valued. In this paper, we show that if G is a direct product of finitely many rational Frobenius groups then every rationally represented character of G is a generalized permutation character. Also we show that the same assertion holds when G is a solvable rational group with a Sylow 2-subgroup isomorphic to the dihedral group of order 8 and an abelian normal Sylow 3-subgroup.     

The integral Novikov conjectures for S-arithmetic groups I

We prove the integral Novikov conjecture for torsion free S-arithmetic subgroups Γ of linear reductive algebraic groups G of rank 0 over a global field k. They form a natural class of groups and are in general not discrete subgroups of Lie groups with finitely many connected components. Since many natural S-arithmetic subgroups contain torsion elements, we also prove a generalized integral Novikov conjecture for S-arithmetic subgroups of such algebraic groups, which contain torsion elements. These S-arithmetic subgroups also provide a natural class of groups with cofinite universal spaces for proper actions. Partially Supported by NSF grants DMS 0405884 and 0604878.     

Picard groups of multiplicative invariants

Let S = kA denote the group algebra of a finitely generated free abelian group A over the field k and let G be a finite subgroup of GL(A). Then G acts on S by means of the unique extension of the natural GL(A)-action on A. We determine the Picard group Pic R of the algebra of invariants R = S ^G . As an application, we produce new polycyclic group algebras with nontrivial torsion in K ₀ . Received: April 25, 1996     

A generalization of 2-Baer groups

A group in which all cyclic subgroups are 2-subnormal is called a 2-Baer group. The topic of this paper are generalized 2-Baer groups, i.e., groups in which the non-2-subnormal cyclic subgroups generate a proper subgroup of the group. If this subgroup is non-trivial, the group is called a generalized T₂-group. In particular, we provide structure results for such groups, investigate their nilpotency class, and construct examples of finite p-groups which are generalized T₂-groups.     

Partitions of groups into absolutely dense subsets

It is proved that any infinite Abelian group with finitely many elements of order two can be partitioned into two subsets that are dense in any nondiscrete group topology, and hence contain no cosets of infinite subgroups. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 706–711, May, 2000.     

Extended genus of torsion-free finitely generated nilpotent groups of class two

The extended genus of a nilpotent group N is the set of isomorphism classes of nilpotent groups M, not necessarily finitely generated, such that the p-localizations M _p , N _p are isomorphic for all primes p. In this article, for any torsion-free finitely generated nilpotent group N of nilpotency class 2, the extended genus of N is analyzed by assigning to each of its members a sequence of triads of matrices with rational entries, generalizing the sequential representation which has been exploited elsewhere in the case when N is abelian. This approach allows, among other things, to obtain examples of groups in the ordinary (Mislin) genus of N     

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.     

New properties of lattices in Lie groups

We study finite extension groups of lattices in Lie groups which have finitely many connected components. We show that every non-cocompact Fuchsian group (these are the non-cocompact lattices in $\text{PSL}\text{(2,}\text{R}\text{)}$) has an extension group of finite index which is not isomorphic to a lattice in a Lie group with finitely many connected components. On the other hand we prove that these are, in an appropriate sense, the only lattices in Lie groups which have extension groups of this kind. We also show that an extension group of finite index of a lattice in a Lie group with finitely many connected components has only finitely many conjugacy classes of finite subgroups. To cite this article: F. Grunewald, V. Platonov, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

On r*-sequenceability and symmetric harmoniousness of groups. II

In this paper we investigate symmetric harmoniousness of groups and connections of this concept to the R*-sequenceability of groups. We prove that, under suitable assumptions, the direct product of a symmetric harmonious group with a group that is R*-sequenceable is R*-sequenceable; we discuss the symmetric harmoniousness of abelian and of nilpotent groups; we also prove that, for a fixed odd prime p, all but possibly finitely many of the nonabelian groups of order pq (q prime, q ≡ 1 (mod p)) are symmetric harmonious. © 1995 John Wiley & Sons, Inc.     

Automorphism groups of some algebras

The automorphism groups of algebras are found in many papers. Using auto-invariance, we find the automorphism groups of the Laurent extension of the polynomial ring and the quantum n-plane (respectively, twisting polynomial ring) in this work. As an application of the results of this work, we can find the automorphism group of a twisting algebra. We define a generalized Weyl algebra and show that the generalized Weyl algebra is simple. We also find the automorphism group of a generalized Weyl algebra. We show that the generalized Weyl algebra A _m,m+n is the universal enveloping algebra of the generalized Witt algebra W(m,m + n). This work was supported by 2007 Research fund of Hanyang University     

Braid groups in genetic code

For every genetic code with finitely many generators and at most one relation, a braid group is introduced. The construction presented includes the braid group of a plane, braid groups of closed oriented surfaces, Artin— Brieskorn braid groups of series B, and allows us to study all of these groups from a unified standpoint. We clarify how braid groups in genetic code are structured, construct words in the normal form, look at torsion, and compute width of verbal subgroups. It is also stated that the system of defining relations for a braid group in two-dimensional manifolds presented in a paper by Scott is inconsistent. Supported by RFBR grant No. 02-01-01118. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 131–158, March–April, 2006.     

Virtually Nilpotent Groups with (almost) All Localizations Trivial

A group G is generically trivial if and only if, for all prime numbers p the localization of G with respect to p is trivial. Taking off from a theorem of Casacuberta and Castellet , we prove that a virtually nilpotent group E is generically trivial if and only if E is perfect. Inspired by this result, we introduce the concept of almost generically trivial groups. Those are groups G such that, for only finitely many primes p the localization of G with respect to p is not trivial. We prove that a virtually nilpotent group E with finitely generated abelianization is almost generically trivial if and only if the abelianization of E is finite.     

Associative nil-algebras and Golod groups

Non-nilpotent, finitely generated, associative nil-algebras are studied as well as their adjoint groups and Golod groups. Solutions are given to some problems in residually finite group theory, questions posed in the Kourovka Notebook included. Supported by RFBR grant No. 03-01-00356. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 231–238, March–April, 2006.     

Characterization of infinite Chernikov groups

It is shown that an infinite locally finite group is a Chernikov group if and only if its Cartesian square G × G contains a subgroup T of finite index such that Aut T possesses the four group V as subgroup with Chernikov centralizer C_T(v) and the centralizers of involutions in V are weakly isolated in T.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 5, pp. 674–677, May, 1990.     

A characterization of alternating groups. II

Let G be a group. A subset X of G is called an A-subset if X consists of elements of order 3, X is invariant in G, and every two non-commuting members of X generate a subgroup isomorphic to A₄ or to A₅. Let X be the A-subset of G. Define a non-oriented graph Γ(X) with vertex set X in which two vertices are adjacent iff they generate a subgroup isomorphic to A₄. Theorem 1 states the following. Let X be a non-empty A-subset of G. (1) Suppose that C is a connected component of Γ(X) and H = 〈C〉. If H ∩ X does not contain a pair of elements generating a subgroup isomorphic to A₅ then H contains a normal elementary Abelian 2-subgroup of index 3 and a subgroup of order 3 which coincides with its centralizer in H. In the opposite case, H is isomorphic to the alternating group A(I) for some (possibly infinite) set I, |I| ≥ 5. (2) The subgroup 〈X^G〉 is a direct product of subgroups 〈C _α〉-generated by some connected components C _α of Γ(X). Theorem 2 asserts the following. Let G be a group and X⊆G be a non-empty G-invariant set of elements of order 5 such that every two non-commuting members of X generate a subgroup isomorphic to A₅. Then 〈XG〉 is a direct product of groups each of which either is isomorphic to A₅ or is cyclic of order 5. Supported by RFBR grant No. 05-01-00797; FP “Universities of Russia,” grant No. UR.04.01.028; RF Ministry of Education Developmental Program for Scientific Potential of the Higher School of Learning, project No. 511; Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 203–214, March–April, 2006.     

Almost isomorphism of Abelian groups and determinability of Abelian groups by their subgroups

An Abelian group A is called correct if for any Abelian group B isomorphisms A ≅ B′ and B ≅ A′, where A′ and B′ are subgroups of the groups A and B, respectively, imply the isomorphism A ≅ B. We say that a group A is determined by its subgroups (its proper subgroups) if for any group B the existence of a bijection between the sets of all subgroups (all proper subgroups) of groups A and B such that corresponding subgroups are isomorphic implies A ≅ B. In this paper, connections between the correctness of Abelian groups and their determinability by their subgroups (their proper subgroups) are established. Certain criteria of determinability of direct sums of cyclic groups by their subgroups and their proper subgroups, as well as a criterion of correctness of such groups, are obtained. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 21–36, 2003.     

Completely torsion-free,finite-rank,almost decomposable groups with torsion factor

This paper deals with almost completely decomposable finite-rank groups G that have rank-1 summands of pairwise noncomparable types. It is well known that every such group has a unique complete quasi-decomposition A with respect to equality. We consider the number of almost completely decomposable groups G with a given quasi-decomposition A for which G/A is isomorphic to ℤ(p ^m). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 3, pp. 61–67, 2007.     

Properly 3-realizable groups

A finitely presented group G is said to be properly 3-realizable if there exists a compact 2-polyhedron K with π₁ (K) ≅ G whose universal cover has the proper homotopy type of a 3-manifold (with boundary). We discuss the behavior of this property with respect to amalgamated products, HNN-extensions, and direct products, as well as the independence with respect to the chosen 2-polyhedron. We also exhibit certain classes of groups satisfying this property: finitely generated Abelian groups, (classical) hyperbolic groups, and one-relator groups. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 95–103, 2005.