Distance magic graphs [formula omitted

We study properties of a group, abelian group, ring, or monoid $B$ which (a) guarantee that every homomorphism from an infinite direct product $\prod_I A_i$ of objects of the same sort onto $B$ factors through the direct product of finitely many ultraproducts of the $A_i$ (possibly after composition with the natural map $B\to B/Z(B)$ or some variant), and/or (b) guarantee that when a map does so factor (and the index set has reasonable cardinality), the ultrafilters involved must be principal. A number of open questions, and topics for further investigation, are noted.

Let $$G_{2}$$ be a group which acts trivially on an abelian group $$G_{1}$$ . According to the Schreier’s Theorem, each 2-cocycle $$\varepsilon \in Z^{2}(G_{2},G_{1})$$ determines a group $$G_{\varepsilon }$$ which is a central extension of $$G_{1}$$ by $$G_{2}$$ , and we will denote this group by $$G_{1}\underset{\varepsilon }{\times }G_{2} $$ and call it the perturbed direct product of $$G_{1}$$ by $$G_{2}$$ under $$\varepsilon $$ . The aim of this paper is to study properties of the perturbed direct products. For two distinct 2-cocycles $$\varepsilon _{1}$$ and $$\varepsilon _{2}$$ , we find necessary and sufficient conditions for $$G_{1}\underset{\varepsilon _{1}}{\times }G_{2} $$ to be isomorphic to $$G_{1}\underset{\varepsilon _{2}}{\times }G_{2}$$ . Furthermore, we obtain some results about decompositions for a given perturbed direct product $$G_{1}\underset{ \varepsilon }{\times }G_{2}$$ when $$G_{1}$$ or $$G_{2}$$ is a nontrivial direct product.

In this chapter we develop the algebra underlying multirate filter structures over free abelian groups of finite rank. Suppose A is a free abelian group of rank N. Each subgroup Δ of A is a free abelian group of rank n ≤ N. The basis theorem describes how Δ sits in A in the sense that it characterizes the quotient group A/Δ. This quotient group can be a finite abelian group a free abelian group of finite rank the direct product of a finite abelian group and a free abelian group of finite rank.

Given a graph G with n vertices and m edges and an abelian group A of order equal to max(n,m). By labeling of A∗-vertex magic labeling and A∗-edge magic labeling we mean a pair one-to-one maps, one from V(G) to A, and another one from E(G) to A, rather than the bijection maps considered in the literature from V(G)∪E(G) to A, and new results were obtained.

Let $G=(V, E)$ be a graph and let $(A,+)$ be an Abelian group with identity elemento. Let $f: V \rightarrow A$ be a vertex labeling and $f^*: E \rightarrow A$ be the induced labeling of $f$, defined by $f^*\left(v_1 v_2\right)=f\left(v_1\right)+f\left(v_2\right)$ for all $v_1 v_2 \in E$. Then $f^*$ again induces a labeling say $f^{* *}: V \rightarrow A$ defined by $f^{* *}(v)=\sum_{v v_1 \in E} f^*\left(v v_1\right)$. A graph $G=(V, E)$ is said to be an Induced $A$-Magic Graph (IAMG) if there exists a non zero labeling $f: V \rightarrow A$ such that $f \equiv f^{* *}$. The function $f$, so obtained is called an Induced $A$-Magic Labeling (IAML) of $G$ and a graph which has no such Induced Magic Labeling is called a Non-induced magic graph. In this paper we discuss the existence of Induced Magic Labeling of some special graphs like $P_n, C_n, K_n$ and $K_{m, n}$.

Let G be a connected complex Lie group. Then there exists the smallest closed complex subgroup G0 of G such that G/G0 is a Stein group (Morimoto). Moreover G0 is a connected abelian Lie group and every holomorphic function on G0 is a constant. G0 is called an (H, C)-group or a toroidal group. Every connected complex abelian Lie group is isomorphic to the direct product G0 × Cm × C*n, where G0 is an (H,C)-group.

We say that a finite group $G$ is conjugacy expansive if for any normal subset $S$ and any conjugacy class $C$ of $G$ the normal set $SC$ consists of at least as many conjugacy classes of $G$ as $S$ does. Halasi, Mar'oti, Sidki, Bezerra have shown that a group is conjugacy expansive if and only if it is a direct product of conjugacy expansive simple or abelian groups. By considering a character analogue of the above, we say that a finite group $G$ is character expansive if for any complex character $alpha$ and irreducible character $chi$ of $G$ the character $alpha chi$ has at least as many irreducible constituents, counting without multiplicity, as $alpha$ does. In this paper we take some initial steps in determining character expansive groups.

In the algebra there are several kinds of structure theorems which can be formulated without operations, using only homological tools. For instance, the wellknown fact that any universal algebra can be subdirectly embedded in a direct product of subdirectly irreducible algebras, can be formulated in a pure categorytheoretical manner. Now the question arises what its dual statement asserts. Our purpose is to give such a category which satisfies certain selfdual conditions, and making use of these, to prove structure theorems and their dual statements. The structure theorems themselves are, of course, well-known statements for algebraic stuctures. However, their duals yield some theorems of unusual type. About the possibility of the dualization there occurs some trouble. The most of the difficulties is at finding selfdual conditions being necessary to prove the theorems. So we must not make use of the condition 'every epimorphism is a normal one' which is fulfilled for groups, since its dual is false. Further the lattice of all congruencerelations of any universal algebra is a so-called compactly generated lattice. This fact plays a very important role in the proof of the theorem according to subdirect embeddings of universal algebras, nevertheless compactly generating is not a se!fdual notion. Applying the theorems proved for certain categories, we establish some particular theorems for rings, groups, modules, respectively. In w 2 we give a detailed enumeration of the usual notions and assertions of the theory of categories with respect to the importance of the dual notions and assertions, moreover, we form a system of selfdual conditions which will be satisfied by the category we are dealing with. w 3 is devoted to the investigation of subdirect embeddings, subdirect irreducibility and to the dualization of those. In w 4 we applicate the results developed before for rings, groups, module ~ and abelian groups. Most of the applications are concerned with rings.

Motivated by the possibility that physics may be effectively five-dimensional over some range of distance scales, we study the possible gaugings of five-dimensional N=2 supergravity. Using a constructive approach, we derive the conditions that must be satisfied by the scalar fields in the vector, tensor and hypermultiplets if a given global symmetry is to be gaugeable. We classify all those theories that admit the gauging of a compact group that is either Abelian or semi-simple, or a direct product of a semi-simple and an Abelian group. In the absence of tensor multiplets, either the gauge group must be semi-simple or the Abelian part has to be U(1)_R and/or an Abelian isometry of the hyperscalar manifold. On the other hand, in the presence of tensor multiplets the gauge group cannot be semi-simple. As an illustrative exercise, we show how the Standard Model SU(3) X SU(2) X U(1) group may be gauged in five-dimensional N=2 supergravity. We also show how previous special results may be recovered within our general formalism.

Computable topological abelian groups

The paper deals with decomposition of a finite abelian group into a direct product of subsets. A family of subsets, the so-called uniquely complemented subsets, is singled out. It will be shown that if a finite abelian group is a direct product of uniquely complemented subsets, then at least one of the factors must be a subgroup. This generalizes Hajos's factorization theorem.

Let π be a set of primes. We will call a group π-separated if it can be decomposed as a central product of a π-torsion group of bounded exponent and a π-radicable group. It is easy to see that an abelian π-separated group can in fact be decomposed as a direct product of bounded and π-radicable factors (Lemma 1.1 below). A poly-π-separated group is one which can be obtained from π-separated groups by forming a finite series of group extensions. We will show here:Theorem 1. Every poly-π-separated nilpotent group is π-separated.The need for such a result arises in model theory in the case that π is the set of all primes, in which case we refer simply to separated and poly-separated groups. In connection with the conjecture that ω-stable simple groups are algebraic, it is useful to have a structure theory for solvable ω-stable groups analogous to the theory available in the algebraic case over algebraically closed fields. In particular the structure of nilpotent ω-stable groups is of interest, for example in connection with the known result [Zi1], [Ne] that the derived subgroup of a connected solvable ω-stable group of finite Morley rank is nilpotent.As a model-theoretic application of Theorem 1 we obtain:Theorem 2. If G is an ω-stable nilpotent group then G may be decomposed as a central product B * D with B and D 0-definable subgroups, B torsion of bounded exponent, and D radicable. In particular, B and D are ω-stable. Furthermore, ω · rk(B ∩ D) ≤ rk D.Corollary. The ω-stable groups of finite Morley rank are exactly the central products B * D of ω-stable nilpotent groups of finite Morley rank with B torsion of bounded exponent, D radicable, and B ∩ D finite.In addition to the purely algebraic Theorem 1, these results depend on Macintyre's characterization [Mac] of the ω-stable abelian groups as exactly the separated ones.

This chapter is devoted to algebraic systems of one binary operation. Groups have been used many times before in the text, as permutations, as modules (a commutative group with an exterior ring multiplication), and elsewhere, but here we organize the systems of one binary operation into one chapter and give some additional specialized results. The most general algebraic system we consider in this chapter is the monoid, a set with one associative binary operation. Some special monoids discussed are unitary monoids, cancellative monoids, and groups. We discuss subsystems, quotient systems, morphisms, and the fundamental morphism theorem. In Sections 9.4 and 9.5 we study more specialized results available for groups alone. We study cyclic groups and connect the order of an element with this concept. Several topics such as center, normalizer, conjugacy classes are all organized around inner automorphisms (φ a (x) = a-xa). We then apply these results to answer some questions about elements of prime order. We offer some standard theorems relating direct products and Cartesian products of groups. We define simple groups and solvable groups. We give an inductive proof of the fundamental theory of Abelian groups which is subsumed in Chapter 10 by the direct sum resolution of a finitely generated module.KeywordsCommutative GroupNormal SubgroupConjugacy ClassSimple GroupCyclic GroupThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Group distance magic and antimagic graphs

A $Γ$\emph{-distance magic labeling} of a graph $G = (V, E)$ with $|V| = n$ is a bijection $\ell$ from $V$ to an Abelian group $Γ$ of order $n$, for which there exists $μ\in Γ$, such that the weight $w(x) =\sum_{y\in N(x)}\ell(y)$ of every vertex $x \in V$ is equal to $μ$. In this case, the element $μ$ is called the \emph{magic constant of} $G$. A graph $G$ is called a \emph{group distance magic} if there exists a $Γ$-distance magic labeling of $G$ for every Abelian group $Γ$ of order $n$. In this paper, we focused on cubic $Γ$-distance magic graphs as well as some properties of such graphs.