Isomorphisms,definable relations,and Scott families for integral domains and commutative semigroups

In the present article, we prove the following four assertions: (1) For every computable successor ordinal α, there exists a Δ _α ⁰ -categorical integral domain (commutative semigroup) which is not relatively Δ _α ⁰ -categorical (i.e., no formally Σ _α ⁰ Scott family exists for such a structure). (2) For every computable successor ordinal α, there exists an intrinsically Σ _α ⁰ -relation on the universe of a computable integral domain (commutative semigroup) which is not a relatively intrinsically Σ _α ⁰ -relation. (3) For every computable successor ordinal α and finite n, there exists an integral domain (commutative semigroup) whose Δ _α ⁰ -dimension is equal to n. (4) For every computable successor ordinal α, there exists an integral domain (commutative semigroup) with presentations only in the degrees of sets X such that Δ _α ⁰ (X) is not Δ _α ⁰ . In particular, for every finite n, there exists an integral domain (commutative semigroup) with presentations only in the degrees that are not n-low.     

An autostable 2 nilpotent group with no Scott family of finitary formulas

We construct an autostable 2 nilpotent group with no Scott family of finitary formulas but having a unique constructivization up to autoequivalence.     

On constructive nilpotent groups

We prove the following: (1) a torsion-free class 2 nilpotent group is constructivizable if and only if it is isomorphic to the extension of some constructive abelian group included in the center of the group by some constructive torsion-free abelian group and some recursive system of factors; (2) a constructivizable torsion-free class 2 nilpotent group whose commutant has finite rank is orderably constructivizable.     

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.     

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.     

On Coleman automorphisms of finite nilpotent groups by cyclic groups

Let G be a finite group and let N be a nilpotent normal subgroup of G such that G/N is cyclic. It is shown that under some conditions all Coleman automorphisms of G are inner. Interest in such automorphisms arose from the study of the normalizer problem for integral group rings.     

On the lower central series of the free nilpotent groups of finite rank

In their article, “On the derived subgroup of the free nilpotent groups of finite rank” R. D. Blyth, P. Moravec, and R. F. Morse describe the structure of the derived subgroup of a free nilpotent group of finite rank n as a direct product of a nonabelian group and a free abelian group, each with a minimal generating set of cardinality that is a given function of n. They apply this result to computing the nonabelian tensor squares of the free nilpotent groups of finite rank. We generalize their main result to investigate the structure of the other terms of the lower central series of a free nilpotent group of finite rank, each again described as a direct product of a nonabelian group and a free abelian group. In order to compute the ranks of the free abelian components and the size of minimal generating sets for the nonabelian components we introduce what we call weight partitions.     

Fuzzy subgroups of nilpotent groups

In 1991, Asaad has suggested the following question: What can be said about the structure of a group G when information is known about the structure of a fuzzy subgroup A of G and vice-versa? This question has been partially investigated in Fuzzy Sets and Systems39 (1991) 323–328. The purpose of this paper is to continue more investigations for the above mentioned question.     

Decompositions of finitely generated nilpotent groups of class 2

In this paper we prove that rational indecomposability is a genus property for finitely generated torsion-free nilpotent groups of class 2. We use this result to determine the genus of finitely generated torsion-free nilpotent groups of class 2 which decompose as a direct product of rationally indecomposable groups. Received: 3 November 2005     

Algebraic sets and coordinate groups for a free nilpotent group of nilpotency class 2

We give a complete classification of the algebraic sets and coordinate groups for the systems of equations in one variable over a free nilpotent group.     

Locally nilpotent linear groups with restrictions on their subgroups of infinite central dimension

Let F be a field and V a vector space over F. If G is a subgroup of GL(V, F), then we define the central dimension of G (denoted by centdim _F G) as the F-dimension of the factor-space V/C _V (G). In this paper, we continue the study of locally nilpotent linear groups satisfying the weak minimal or the weak maximal condition on their subgroups of infinite central dimension started in Kurdachenko et al. (Publ Mat 52:151–169, 2008). Supported by Proyecto MTM2007-60994 of Dirección General de Investigación MEC (Spain).     

Signature of relations in mapping class groups and non-holomorphic Lefschetz fibrations

We introduce the notion of signature for relations in mapping class groups and show that the signature of a Lefschetz fibration over the 2-sphere is the sum of the signatures for basic relations contained in its monodromy. Combining explicit calculations of the signature cocycle with a technique of substituting positive relations, we give some new examples of non-holomorphic Lefschetz fibrations of genus and which violate slope bounds for non-hyperelliptic fibrations on algebraic surfaces of general type.

     

Definable nilpotent and soluble envelopes in groups without the independence property

We prove that in a group without the independence property a nilpotent subgroup is always contained in a definable nilpotent subgroup of the same nilpotency class. The analogue for the soluble case is also shown when the subgroup is normal in the ambient group.     

Description of finite nilpotent groups of the 2nd degree with a prime odd period

We describe the mentioned groups, assigning values to centralizers of the generating elements and defining a group operation on corteges of elements of the Galois field with a prime odd characteristic. On the indicated groups we define odules over the Galois field and describe these odules. We prove that the considered groups are those with unique extraction of root.     

Classification and discrete cocompact subgroups of some 7-dimensional connected nilpotent groups

Summary For real connected nilpotent groups, 7 is the lowest dimension where there are infinitely many non-isomorphic groups, and also where some groups (indeed, uncountably many) have no discrete cocompact subgroups. In [21] one infinite family <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>\mathcal{G}$ of 7-dimensional groups was identified and classified. Discrete cocompact subgroups H were identified for some groups in $\mathcal{G}$ in [10], along with simple quotients of $C^{*}(\mathrm{H})$ and relevant flows $(\mathrm{H}_3,\mathbf{T}^3)$. In this paper, such H and attributes are determined for more groups in $\mathcal{G}$; in particular, the members of $\mathcal{G}$ that admit discrete cocompact subgroups are identified precisely. In achieving some of these results, we consider other known ways of classifying the groups in $\mathcal{G}$, and also the classification of the analogous family of complex groups.     

On the Witten-Reshetikhin-Turaev representations of mapping class groups

We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The Witten-Reshetikhin-Turaev TQFTs associated to and induce linear representations of this group. We show that the denominators of matrices which describe these representations over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus with no colored points, the representations have finite image.

     

Finite groups with 2-nilpotent subgroups of even index

Finite groups with 2-nilpotent subgroups of even index are described. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 700–705, May, 1997. Translated by A. I. Shtern     

Ideal class groups and their subgroups of real quadratic fields

A necessary and sufficient condition is given for the ideal class group H(m) of a real quadratic field Q (√m) to contain a cyclic subgroup of ordern. Some criteria satisfying the condition are also obtained. And eight types of such fields are proved to have this property, e.g. fields withm=(z ⁿ +t−1)²+4t(witht|z ⁿ −1), which contains the well-known fields withm=4z ⁿ +1 andm=4z ²ⁿ +4 as special cases. Project supported by the National Natural Science Foundation of China.