Factorization Of Abelian Groups Research Articles

Nonsingular splittings of cyclic groups

In this paper, we study nonsingular splittings of cyclic groups. We introduce two new notations “direct logarithm” and” special direct KM-logarithm”. Using Kummer and Mills' results on characters and a result from the factorization of abelian groups, we prove that if there is a prime q such that M splits Zq, then there are infinitely many primes p such that M splits Zp. We also obtain some results on the existence and nonexistence of split [−k1,k2]⁎ sets, and give a characterization of splitter sets B such that [−1,5]⁎ splits Zp.

Abelian group factorization from cyclic and quasi-cyclic codes

In this paper, we use the algebraic structures of cyclic codes and algorithmic techniques to find factorizations of abelian groups from cyclic codes. We construct specific subclasses of quasi-cyclic codes and provide the conditions with which we obtain a normalized factorization of certain abelian groups. The factorization, in both cases, is constituted by two sets, one corresponding to the cyclic code and the other corresponding to the words that represent all possible error polynomials of the cyclic code besides the zero vector.

Hausdorffization and Such

If G is a group, with commutator subgroup G′, then the quotient group G/G′ serves as an “Abelianization” of G. Any homomorphism from G to an Abelian group factors through G/G′. Similarly, any topological space has a “Hausdorffization,” as well as several more such constructs (which is the subject of this paper).

On the factorization of Abelian groups

In this note we consider a conjecture made by Professor de Bruijn and subsequent generalizations of this conjecture. We show by using constructions of his that in a certain sense these generalizations are the best possible. A survey of these known results is presented. Finally some new results of this kind are given.

A simplified scheme for GUT-inspired theories with multiple Abelian factors

Grand Unified Theories often involve additional Abelian group factors apart from the standard model hypercharge, that generally lead to loop-induced mixing gauge kinetic terms. In this letter, we show that at the one-loop level this effect can be avoided in many cases by a suitable choice of basis in group space and present a general scheme for the construction of this basis. In supersymmetric theories however, a residual mixing in the soft SUSY breaking gaugino mass terms may appear. We generalize the renormalization group equations for the gaugino mass terms to account for this effect. In a further calculation we also present the necessary adjustments in the renormalization group equations of the trilinear soft breaking couplings and the soft breaking scalar mass squares.

Aperiodic logarithmic signatures

In this paper we propose a method to construct logarithmic signatures which are not amalgamated transversal and further do not even have a periodic block. The latter property was crucial for the successful attack on the system MST 3 by Blackburn, Cid and Mullan (2009). The idea for our construction is based on the theory in Szabó's book “Topics in Factorization of Abelian Groups”.

An Application of Cyclotomic Polynomial to Factorization of Abelian Groups

If a finite abelian group G is a direct product of its subsets such that G = A1···Ai···An, G is said to have the Hajos-n-proprty if it follows that one of these subsets, say Ai is periodic, meaning that there exists a nonidentity element g in G such that gAi = Ai . Using some properties of cyclotomic polynomials, we will show that the cyclic groups of orders pα and groups of type (p2,q2) and (pα,pβ) where p and q are distinct primes and α, β integers ≥ 1 have this property.

A Paper-and-Pencil gcd Algorithm for Gaussian Integers

Sandor Szab6 (sszabo7@hotmail.com) was born and grew up in Hungary and graduated from E6tv6s University in Budapest. He has taught at the Technical University in Budapest, has held a variety of visiting positions, and currently is on the faculty of the University of Pecs in Hungary. His main mathematical interests are in algebra and its applications. He has written two monographs, Algebra and Tilingq with S. K. Stein, and Topics in Factorization of Abelian Groups.

Replacement of Factors By Subgroups in the Factorization of Abelian Groups

Replacement of Factors By Subgroups in the Factorization of Abelian Groups

Flat directions in three-generation free-fermionic string models

In quasi-realistic string models that contain an anomalous U(1) the non-zero Fayet-Iliopoulos term triggers the shifting of the original vacuum to a new one along some flat direction, so that SUSY is preserved but the gauge group is partially broken. The phenomenological study of these models thus requires as a first step the mapping of the space of flat directions. We investigate F- and D-flat directions in several three-generation SU(3) C × SU(2) L × U(1) Y free-fermionic string models and discuss the typical scenarios that generically arise. When they exist, we systematically construct the flat directions that preserve hypercharge, only break abelian group factors, and can be proven to remain F-flat to all orders in the non-renormalizable superpotential.

A dihedral calculus for groups

The classical method of abelianization is an important tool in combinatorial group theory. It is an algorithm for tbe calculation of the commutator factor group and thus gives a survey of the abelian homomorphic images of a group, since any homomorphism of a group into an abelian group factors through the commutator factor group. A non-commutative analogue of abelianization can be developped if one replaces the abelian groups by the (generalized) dihedral groups; in other words, there exists a method which gives a survey of the (generalized) dihedral homomorphic images of a group. The foundations of this new method will be given in Sections 1 and 2. The most conspicuous difference between abelianization and "dihedralization" is the fact that there is no single substitute for the commutator subgroup; it will be shown that the commutator subgroup has to be replaced by a certain canonical set of normal subgroups. Subsequently, instead of having a single algorithm as in the abelian case, several algorithms have to be carried out. The algorithms, which are described in Section 3, are not more complicated than the corresponding one of abelianization. Notation, preliminaries. X- Y denotes the set difference; if Y consists of a single element y, we write X - y. IX[ denotes the cardinal of the set X. If S is a subset of the group G, (S) denotes the subgroup generated by the set S, and S- 1 denotes the set of all inverse elements of S. If G and H are groups, H < G means that H is a subgroup of G. IfH is a normal subgroup of G and x, y are elements of G such that xyN = yxN, we will occasionally say that x and y commute rood N. An elementary abelian 2-group is a group which contains only elements of order at most two. If G is a group, the subgroup generated by all squares is denoted by G 2. Clearly, G 2 iS the smallest of all normal subgroups which have an elementary abelian 2-group as factor group.

THE FACTORIZATION OF ABELIAN GROUPS (II)

Journal Article THE FACTORIZATION OF ABELIAN GROUPS (II) Get access A. D. SANDS A. D. SANDS Dundee Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 13, Issue 1, 1962, Pages 45–54, https://doi.org/10.1093/qmath/13.1.45 Published: 01 January 1962 Article history Received: 29 May 1961 Published: 01 January 1962