The generalization of one classical Smel’kin’s theorem for polycyclic groups is obtained. A. L. Smelkin proved that if G is a polycyclic group, then it is a virtually residually finite p-group for any prime p. Recall that a group G is said to be a residually finite p-group if for every nonidentity element a of G there exists a homomorphism of the group G onto some finite p-group such that the image of the element a differs from 1. A group G will be said to be a virtually residually finite p-group if it contains a finite index subgroup which is a residually finite p-group. One of the generalizations of the notation of polycyclic group is a notation of soluble finite rank group. Recall that a group G is said to be a group of finite rank if there exists a positive integer r such that every finitely generated subgroup in G is generated by at most r elements. For soluble groups of finite rank the following necessary and sufficient condition to be a residually finite π-group for some finite set π of primes is obtained. If G is a group of finite rank, then the group G is a residually finite π- group for some finite set π of primes if and only if G is a reduced poly-(cyclic, quasicyclic, or rational) group. Recall that a group G is said to be a reduced group if it has no nonidentity radicable subgroups. A group H is said to be a radicable group if every element h in H is an mth power of an element of H for every positive number m. It is proved that if a soluble group of finite rank is a residually finite π- group for some finite set π of primes, then it is a virtually residually finite nilpotent π-group. We prove also the following generalization of Smel’kin’s theorem. Let π be a finite set of primes. If G is a soluble group of finite rank, then the group G is a virtually residually finite π-group if and only if G is a reduced poly-(cyclic, quasicyclic, or rational) group and G has no π-radicable elements of infinite order. Recall that an element g in G is said to be π-radicable if g is an mth power of an element of G for every positive π-number m.

 

There is given a survey of results on a structure of semisimple finite dimensional Hopf algebras with isomorphic irreducible non-one-dimensional representations. There is found an explicit form of group-like elements in these Hopf algebras and a structure of the dual a Hopf algebra.

Let G be finitely generated Artin group with tree-structure defined by the presentation G =< a1, ..., an; < aiaj >mij=< ajai >mji , i, j = 1, n >, where mij is number that corresponds to symmetrical matrix of Coxeter, and mij > 2, i 6= j a group G matches the end coherent tree-graph Γ such that if the tops of some edge e of the graph Γ match the form ai and aj , then the edge e corresponds to the ratio of the species < aiaj >mij=< ajai >mji . Artin groups with a tree-structure was introduced by V. N. Bezverkhnii, theirs algorithmic problems were considered by V. N. Bezverkhnii and O. Y. Platonova (Karpova). The group G can be represented as the tree product 2-generated of the groups, united by a cyclic subgroups. We proceed from the graph Γ of the group G to the graph Γ the following follows: the tops of some edge e of the graph Γ put in correspondence Artin groups the two forming Gij =< ai , aj ; < aiaj >mij=< ajai >mji> and Gjk =< aj , ak; < ajak >mjk=< akaj >mkj>, and edge e will match cyclic subgroup < aj >. This paper considers the theorem on the freedom of the Artin groups with a tree-structure: let H be finitely generated subgroup of an Artin group G with a tree-structure, while for any g ∈ G and every subgroup Gij , i 6= j, executed equality gHg−1 ∩ Gij = E then H is free. In the proof of use of the ideas V. N. Bezverkhnii on bringing many forming of the subgroup to a special set.

 

In the work positively with the problem of the conjugation words in a tree product of free groups associated with cyclic subgroups. This work is a generalization of the well-known results of Lipschutz S. for a free product of two free groups with cyclic amalgamation.When solving the main problem it is proved the solvability of the problem of intersection of finite generated subgroup of this group with a cyclic subgroup of the factor group and the problem of the intersection of the co-set of finite generated by subgroup with a cyclic subgroup of the factor group.

 

 

A set of binary relations closed with respect to some collection of operations on relations forms an algebra called an algebra of relations. The first mathematician who treated algebras of relations from the point of view of universal algebra was Alfred Tarski. In the investigation of algebras of relations, one of the most important directions is the study of those of their properties which can be expressed by identities. This leads us to the consideration of varieties generated by classes of algebras of relations. For any set Ω of operations on binary relations, let R{Ω} denote the class of all algebras isomprphic to ones whose elements are binary relations and whose operations are members of Ω. Let V ar{Ω} be the variety generated by R{Ω}. As a rule, operations on relations are defined by formulas of the firstorder predicate calculus. These operations are called logical. One of the most important classes of logical operations on relations is the class of Diophantine operations (in other terminology – primitive-positive operations). An operation on relations is called Diophantine if it can be defined by a formula containing in its prenex normal form only existential quantifiers and conjunctions. A Diophantine operation is called atomic if it can be defined by a first order formula containing in its prenex normal form only existential quantifiers. It is clear that such formulas contain only one atomic subformula. Hence atomic operations are unary operations. There exist nine atomic operations (excepting identical). We concentrate our attention on the Diophantine operation of relation product ◦ and on the atomic operation of double cylindrification ∇ that are defined as follows. For any relations ρ and σ on U, put ρ ◦ σ = {(u, v) : (∃w)(u, w) ∈ ρ(w, v) ∈ σ}, ∇(ρ) = {(u, v) : (∃w, z)(w, z) ∈ ρ}. In the paper, the bases of identities for the variety V ar{◦, ∇} is found: an algebra (A, ·, ∗ ) of the type (2, 1) belongs to the variety V ar{◦, ∇} if and only  if it satisfies the identities: (xy)z = x(yz), x ∗∗ = x ∗ , (x ∗ ) 2 = x ∗ , x ∗y ∗ = y ∗x ∗ , x ∗ (xy) ∗ = (xy) ∗y ∗ = (xy) ∗ , (xy∗ z) ∗ = x ∗y ∗ z ∗ = x ∗yz, xyz∗ = xyx∗ z ∗ , x ∗ z = x ∗ z ∗yz.

 

The class algebra and the double coset algebra are two commutative subalgebras of the group algebra of the symmetric group. The connection coefficients of these two algebraic structures received significant attention in combinatorics as they provide the number of factorizations of a given permutation into an ordered product of permutations satisfying given cyclic structures. While they are usually studied separately, these two families of connection coefficients share strong similarities. They are both equal to some sums of characters, respectively the irreducible characters of the symmetric group and the zonal spherical functions, two specific cases of a more general family of characters named Jack’s characters. Jack’s characters are defined as the coefficients in the power sum expansion of the Jack’s symmetric functions, a family of symmetric polynomials indexed by a parameter α. Connection coefficients of the class algebra corresponds to the case α = 1 (Jack’s symmetric functions are proportional to Schur polynomials in this case) and the connection coefficients of the double coset algebra corresponds to the case α = 2 (Jack’s symmetric functions are equal to zonal polynomials). We define Jack’s connection coefficients to provide a unified approach for general parameter α. This paper introduces these generalized coefficients and focus on their computations. More specifically we focus on the generalization of the formula giving the number of factorizations of a permutation of a given cyclic structure into the product of r transpositions. We use the action of the Laplace-Beltrami operator on Jack’s symmetric functions to provide a general formula and make this formula explicit for some given values of r.

 

The connection between a structure of abelian group and a structure of endomorphism ring is a classic question in abelian group theory. In particular, Baer and Kaplansky proved that this connection is very strong for torsion groups: two abelian torsion groups are isomorphic if and only if their endomorphism ring are isomorphic. In more general cases for torsion-free and mixed abelian groups the Baer-Kaplansky theorem is fails. This paper deals with a class of p-local torsion-free abelian of finite rank. Let K be a field such that Q ⊂ K ⊂ Qbp and let R = K ∩ Zbp, where Zbp is the ring of p-adic integers, Qbp is the field of p-adic numbers, Q is the field of rational numbers. We say that K is a splitting field (R is a splitting ring) for a p-local torsion-free reduced group A or that group A is K-decomposable group if A ⊗Zp R is the direct sum of a divisible R-modules and a free R-modules. Torsion-free p-local abelian groups of finite rank with quadratic splitting field K are characterized. As an application it is proved that K-decomposable plocal torsion free abelian groups of finite rank are isomorphic if and only if their endomorphism rings are isomorphic.

 

Definition of n-ary group is obtained from the definition of group by replacement of associative and reversible binary operation on n-ary associative operation, uniquely reversible at each site. In this paper we study the connection between the generating sets n-ary group and the generating sets the group to which reducible given n-ary group, according to Post - Gluskin - Hossu theorem. In the first part of the article describes the process that allows knowing the generating set of the group to which this is reducible n-ary group in accordance with this theorem, find a generating set of the most n-ary group. We prove that if the group hA, ◦ai, obtained by an element a of n-ary group hA, [ ]i in accordance with Post-Gluskin-Hossu theorem, generated by a set M, then n-ary group hA, [ ]i generated by a set M ∪ {a}. n-Ary group hA, [ ]i called derived of group A, if [a1a2 . . . an] = a1a2 . . . an for any a1, a2, . . . , an ∈ A. Found conditions under which generating sets the group and n-ary group, derived of this group, are identical. We prove that the n-ary group hA, [ ]i, derived of group hA, ◦i with identity e and generating set M, is generated by a set M too, if c1 ◦ c2 ◦ . . . ◦ cm(n−1)+1 = e for some c1, c2, . . . , cm(n−1)+1 ∈ M, m > 1. From this we deduce corollary: n-ary group hA, [ ]i, derived of group hA, ◦i finite period m(n−1)+ 1 > 3 with generating set M, is generated by a set M too. In specifically, n-ary group hA, [ ]i, derived of cyclic group hA, ◦i of order m(n − 1) + 1 > 3 is cyclic and is generated by the same element that group hA, ◦i. Are a few examples of finding generating sets for n-ary groups . In the second part we study the inverse problem of finding generators sets of binary groups, if we know the generating sets of n-ary groups from which this binary groups are obtained (according to the Post-Gluskin-Hossu theorem). Proved that the group hA, ◦ai, obtained by an element a of n-ary group hA, [ ]i with generating set M, generated by the set M ∪ {d = [a . . . a | {z } n ]}, if the automorphism β(x) = [axa a . . . a ¯ | {z } n−3 ] of group hA, ◦ai is satisfied Mβ = {[aMa a . . . a ¯ | {z } n−3 ]} ⊆ M. (2) From this we have the corollary: let n-ary group hA, [ ]i generated by a set M, satisfying (2) for some a ∈ M. Then: 1) the group hA, ◦ai generated by the set (M{a}) ∪ {d}; 2) if a – idempotent in hA, [ ]i, then the group hA, ◦ai generated by the set M{a}. At the end of the work described generating sets of binary groups hA, ◦ai, found from the known generating sets of n-ary groups hA, [ ]i with nonempty center Z(A). 

 

Tits proved that for any finitely generated linear group G, the following statement holds: G is either solvable-by-finite, or it contains a subgroup isomorphic to the free group F2 of rank 2. This leads to the concept of the Tits’ alternative for a class of groups: For a class C of groups the Tits’ alternative holds, if an arbitrary group G from this class is either solvable-by-finite, or it contains a subgroup isomorphic to the free group F2 of rank 2. A number of works have addressed the studying of the classes of groups for which the Tits’ alternative holds. The Tits’ alternative is related to the following problem which has been independently studying for a long time in combinatorial group theory: Find the class of groups possessing the following property: for an arbitrary group G from this class, the following alternative holds: either a non-trivial identity holds on the group G, or G contains a subgroup isomorphic to the free group F2 of rank 2. For subgroups of the groups with one defining relation, this problem was fully studied by D. I. Moldavanskii, A. A. Chebotar’, A. Karrass and D. Solitar. For groups satisfying small cancellation conditions, this problem was studied by V. P. Klassen in describing the subgroups of such groups. The full description of Abelian subgroups of arbitrary F-groups is given in the famous monograph by R. Lindon and P. Schupp. In the present work, this result is strengthened: we give a description of subgroups of F-groups, on which a non-trivial identity holds and prove the Tits alternative for subgroups of F-groups. More accurately, we prove that for the subgroups of Fuchsian groups, the strengthened variant of the Tits’ alternative holds: An arbitrary subgroup H of a Fuchsian group either is solvable group of degree 6 3 or alternating group A(5), or H contains a subgroup isomorphic to the free group of rank 2, No non-trivial identity does hold on a subgroup H of an arbitrary Fuchsian group G if and only if H contains a subgroup isomorphic to the free group F2 of rank 2.

 

In the paper, we obtain that a linear combination of the periodic and periodic Hurwitz zeta-functions, and more general combinations of these functions have infinitely many zeros lying in the right-hand side of the critical strip.

 

Lower estimates of the irrationality measure of logarithms of rational numbers considered by many foreign authors: M. Waldschmidt [1], A. Baker and G. W¨ustholz [2], A. Heimonen, T. Matala-aho, K. Va¨a¨na¨nen [3], Q. Wu [4], G. Rhin [5] and P. Toffin [6]. In their works they used various integral constructions, giving small linear forms from logarithms and other numbers, calculated asymptotic of integrals and coefficients of the linear forms using the saddle point method, Laplace theorem, evaluated the denominator coefficients of the linear forms using various schemes "reduction of prime numbers". Review of some methods from the theory of diophantine approximation of logarithms of rational numbers at that time was introduced in 2004 by V. Zudilin [7]. Then V. Kh. Salikhov in [8] considerably improved estimate of the irrationality measure of log 3, based on the same asymptotic methods, but used a new type of integral construction, which has property of symmetry. Subsequently, V. Kh. Salikhov due to usage of already complex symmetrized integral improved estimate of the irrationality measures of π [9]. In the future, this method (as applied to diophantine approximation of logarithms of rational numbers) was developed by his pupils: E. S. Zolotuhina [10, 11], M. Yu. Luchin [12, 13], E. B. Tomashevskaya [14]. It led to improvement estimates of the irrationality measure following numbers: µ(log(5/3)) 6 5.512... [14], µ(log(8/5)) < 5.9897... [12], µ(log(7/5)) 6 4.865... [14], µ(log(9/7)) 6 3.6455... [10], µ(log(7/4)) < 8.1004... [13]. In this paper due to usage the symmetrized real integral we obtain a new estimate of the irrationality measure of τ = log (37/30), µ(τ ) < 65.3358. First time estimate of the irrationality measure of log (37/30) was received in 1993 by A. Heimonen, T. Matala-aho, K. Va¨a¨na¨nen [1]. In their work they received a common criterion that allows to evaluate irrationality measure of numbers of the form log(1 − (r/s)), where r/s ∈ [−1, 1) (r, s ∈ N). As an example, they led a table with the resulting estimates at some values r/s. One of these values was the number r/s = −7/30, which gave following estimate: µ(log (37/30)) 6 619.5803... We also note, that for obtain a new estimate the optimal parameters of integral construction were calculated using the developed by the author of a computer program, which uses the Mathcad calculations.

Let Fn, n > 2 denote the free group generated by n letters x1, . . . , . . . , xn and Aut(Fn) be the automorphism group of Fn. Certain subgroup of the group Aut(Fn) are considered. First of all examine the palindromic automorphism group ПA(Fn). This group first defined Collins in [1], which is related to congruence subgroups of SL(n,Z), and symmetric automorphism group of the free group. It is calculate the center of the palindromic automorphism group. For this used combinatorics on words of the group Fn. Second theme of this paper connect with faithfulness of a linear representation of the group elementary palindromic automorphisms EПA(Fn). It is show that some concrete representation are not linear. For this use the subgroup IA(Fn) of group Aut(Fn) [15].

 

Development of the asymmetric cryptography started with the appearance of the first knapsack information protection system, when, in 1978, Ralph Merkel and Martin Hellman proposed to use different keys for forward and reverse mapping data for encryption. Now this model, like many based on are considered to be insecure. As a result the authority of knapsack systems was low. However, some of these systems are still considered persistent, for example, the model proposed in 1988 by Ben Shore and Ronald Rivest. In the article stated and solved the problem of argumentation of cryptographic strength of the non-standard knapsack information security systems. Justified diophantine difficulties that arise in the study of vulnerabilities of the investigated information security systems. Revealed the qualitative features of non-standard knapsack systems that increase their resistance to known attacks. In this paper, we propose a mathematical model of polyalphabetic cryptosystem, in which the algorithm of inverse transformation of closed text is algorithmically unsolvable problem for the analyst. It’s permeated with the idea K.Shennon, who believed that cryptosystems, containing Diophantine problems, have the greatest variation in the selection of key.

 

Article represents the review of properties of variety left nilpotent of the class not more than 3 Leibniz algebras and its subvarieties. The characteristic of basic field will be equal to zero. A Leibniz algebra is an algebra with multiplication satisfying the Leibniz identity (xy)z = (xz)y + x(yz). In other words, the operator of right multiplication is a derivation of the algebra. Since Leibniz identity equivalent to the Jacobi identity, in case multiplication in Leibniz algebra is anti-commutative, it is obvious that the Leibniz algebras are generalizations of concept of Lie algebtras. The variety 3N is defined by identity x(y(zt)) ≡ 0 possesses some extreme properties (properties, which any its own subvariety possesses, while the variety doesn’t possess them). As the basic field has zero characteristic zero, then any identity is equivalent to the system of multilinear identities, that allows to use well-developed theory of representations of the symmetric group. In addition to using the classical results of the structural theory of rings and linear algebras, representation theory, as well as the structural theory of varieties of associative algebras, and the use of original asymptotic and combinatorial arguments with application identities and Young diagrams allowed to receive the following results: the variety 3N has almost exponential growth, almost polynomial growth of colength, almost finite multiplicity. Moreover, this variety has almost associative type, that is his own cocharacter any subvarieties lies in the hook. In this work are considered also subvarieties of variety 3N: held description of the complete list of varieties with almost polynomial growth; proved integrality of exponents any proper subvariety of variety 3N.

 

Bell‘s numbers B(s) defines the amount partitions of s-element set and with growth s they have an exponentiale growth. That‘s why the asymptotic‘s investigation s >> 1 of sequence {B(s)} of Bell‘s numbers B(s) becomes actual, for example, if do the following combinatorial sum. Let‘s take a discrete space of elementary event containing s points with given law of probability distribution p1; . . . ; ps, p1 + . . . + ps = 1. On configurations of partitions one should define such a partition at which minimum of informational Shanon‘s entropy is gained. One can face with this problem when the optimization of block-control of difficult cybernetic systems is present. In this work some asymptotic properties of sequence of Bell‘s numbers are considered. The main result of work represents the correlation: lims→∞ B(s)B(s + 2) B2(s + 1) = 1, where B(s); B(s + 1); B(s + 2) — Bell‘s numbers with numerals s; s + 1; s + 2. This result shows that asymptotical sequence of Bell‘s numbers behaved themselves geometrical progression with denominator x∗ = B(s + 1)/B(s). In the frames of additive presentation of Bell‘s numbers with the help of Stirling‘s numders the asymptotics is set up B(s) St(s; n∗) (n ∗ ) s/(n ∗ )!, where n∗ = [x∗]. Thus, a new class of sequences is up, the topology of which is characterized by the asymptotics in the form of the geometrical progression. Thus, a new class of sequences is established, the topology of wich is characterized by asymptotics in the form of geometrical progression.

 

Let α ∈ (0; 1) be an irrational number. Study of the distribution of fractional parts {iα} on the interval (0; 1) is a classical question in number theory. In particular, H.Weyl proved that this sequence is uniformity distributed modulo 1. Since this work, various estimates for the remainder term of the asymptotic formula for the number of the sequence of points belonging to a given interval are actively investigated. Another type of problems about considered sequence are problems associated with the famous three lenghts theorem (Steinhaus conjecture), which state that a tiling of the unit interval generated by the points of the sequence, composed of intervals of two or three different lengths. Moreover, in the second case the length of the greatest inteval exactly equals the sum of the lengths of two other intervals. It was find out that the geometry of these tilings is closely connected with the first return maps for circle rotations, Hecke-Kesten problem on bounded remainder sets, combinatorics of Sturmian sequences, dynamics of two-color rotations of the circle , and some other problems. This paper deals to combinatorial properties of the sequence {iα}, such as permutations πα,n, generated by the points {iα}, 1 6 i 6 n. It is proved that there is a one-to-one correspondence between these permutations and the intervals of Farey tilings of the level n. Here Farey tiling of the level n is a tiling of the interval [0; 1] generated by irreducible rational fractions of the form a b with denominator 0 < b 6 n. The proof is based on one theorem of V.T.Sos, that allows to compute the permutation πα,n using only πα,n(1) and πα,n(n). Also we use the fact that the ends of intervals of the Farey coincide with the points of discontinuity of the functions {kα}−{lα}. As an application it is proved that there are exactly 1 + Pn k=2 ϕ(k) different permutations πα,n for any fixed n. Another our result states that the permutation πα,n uniquely determines permutations πα,m with n < m < πα,n(1) + πα,n(n) and does not uniquely determine the permutation πα,m with m = πα,n(1) + πα,n(n).