scholarly journals Galkin Quandles, Pointed Abelian Groups, and Sequence A000712

10.37236/2676 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
William E. Clark ◽  
Xiang-dong Hou
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Knot Invariants ◽  
Group A

For each pointed abelian group $(A,c)$, there is an associated Galkin quandle $G(A,c)$ which is an algebraic structure defined on $\Bbb Z_3\times A$ that can be used to construct knot invariants. It is known that two finite Galkin quandles are isomorphic if and only if their associated pointed abelian groups are isomorphic. In this paper we classify all finite pointed abelian groups. We show that the number of nonisomorphic pointed abelian groups of order $q^n$ ($q$ prime) is $\sum_{0\le m\le n}p(m)p(n-m)$, where $p(m)$ is the number of partitions of integer $m$.

A generalization of co-Hopfian abelian groups

2017 ◽  
Vol 27 (04) ◽  
pp. 351-360
Author(s):  
Andrey Chekhlov ◽  
Peter Danchev
Keyword(s):  
Abelian Group ◽  
Representation Theorem ◽  
Abelian Groups ◽  
Classical Notion ◽  
Group A ◽  
Hopfian Group ◽  
Comprehensive Study

We define the concept of an [Formula: see text]-co-Hopfian abelian group, which is a nontrivial generalization of the classical notion of a co-Hopfian group. A systematic and comprehensive study of these groups is given in very different ways. Specifically, a representation theorem for [Formula: see text]-co-Hopfian groups is established as well as it is shown that there exists an [Formula: see text]-co-Hopfian group which is not co-Hopfian.

Construction of orders in Abelian groups

1961 ◽  
Vol 57 (3) ◽  
pp. 476-482 ◽  
Author(s):  
H.-H. Teh
Keyword(s):  
Abelian Group ◽  
Binary Relation ◽  
Abelian Groups ◽  
Group A

Let G be an Abelian group. A binary relation ≥ denned in G is called an order of G if for each x, y, z ε G,(i) x ≥ y or y ≥ x (and hence x ≥ x);(ii) x ≥ y and y ≥ x ⇒ x = y,(if x ≥ y and x ≠ y, we write x > y);(iii) x ≥ y and y ≥ z ⇒ x = z;(iv) z ≥ y ⇒ x + z ≥ y + z.

scholarly journals A Study on Finite Abelian Groups: Sylow’s Theorems Based

2017 ◽  
Vol 1 (3) ◽  
Author(s):  
Amaira Moaitiq Mohammed Al-Johani
Keyword(s):  
Abelian Group ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Commutative Rings ◽  
Abstract Algebra ◽  
Vector Spaces ◽  
Finite Abelian Groups ◽  
Study Objective ◽  
Group Operation ◽  
Basic Concepts

In abstract algebra, an algebraic structure is a set with one or more finitary operations defined on it that satisfies a list of axioms. Algebraic structures include groups, rings, fields, and lattices, etc. A group is an algebraic structure (????, ∗), which satisfies associative, identity and inverse laws. An Abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, these are the groups that obey the axiom of commutatively. The concept of an Abelian group is one of the first concepts encountered in abstract algebra, from which many other basic concepts, such as rings, commutative rings, modules and vector spaces are developed. This study sheds the light on the structure of the finite abelian groups, basis theorem, Sylow’s theorem and factoring finite abelian groups. In addition, it discusses some properties related to these groups. The researcher followed the exploratory and comparative approaches to achieve the study objective. The study has shown that the theory of Abelian groups is generally simpler than that of their non-abelian counter parts, and finite Abelian groups are very well understood.  

scholarly journals On Injective Sheaves

1964 ◽  
Vol 7 (3) ◽  
pp. 415-423
Author(s):  
H. Kleisli ◽  
Y.C. Wu
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Crucial Point ◽  
Arbitrary Ring ◽  
Injective Modules ◽  
Group A ◽  
Group B ◽  
Divisible Groups ◽  
Group D ◽  
Divisible Abelian Group

A divisible abelian group D can be characterized by the following property: Every homomorphism from an abelian group A to D can be extended to every abelian group B containing A. This together with the result that every abelian group can be embedded in a divisible group is a crucial point in many investigations on abelian groups. It was Baer, [1], who extended this result to modules over an arbitrary ring, replacing divisible groups by injective modules, that is, modules with the property mentioned above. Another proof was found later by Eckmann and Schopf, [3]. This proof assumes the proposition to hold for abelian groups and transfers it in a very simple and elegant manner to modules. In the sequel, we shall refer to this proof as to the Eckmann-Schopf proof.

The order indiscernibles of divisible ordered abelian groups

1984 ◽  
Vol 49 (1) ◽  
pp. 151-160
Author(s):  
David Rosenthal
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Simple Condition ◽  
Algebraic Invariants ◽  
Image Position ◽  
And Algebra ◽  
Simplified Form

There has been much work in developing the interconnections between model theory and algebra. Here we look at a particular example, the divisible ordered abelian groups, and show how the indiscernibles are related to the algebraic structure. Now a divisible ordered abelian group is a model of Th (Q, +, 0, <) and so is linearly ordered by <. Thus the theory is unstable and has a large number of models. It is therefore unrealistic to expect that a simple condition will completely determine a model. Instead we would just like to obtain nice algebraic invariants.Definition. A subset C of a divisible ordered abelian group is a set of (order) indiscernibles iff for every sequence of integers n1,…,nk and for every c1 < … < ck and d1 < … < dk in CNote that this simplified form of indiscernibility is an immediate consequence of quantifier elimination for the theory. The above definition could be formulated in the language of +, 0, < but we have used subtraction as a matter of convenience. Similarly we may also use rational coefficients. Also note that a set of order indiscernibles is usually defined with respect to some external order. But in this case there are only two possibilities: the ordering inherited from or its reverse. So we will always assume that a set of order indiscernibles has the ordering inherited from . We may sometimes refer to a set of indiscernibles as a sequence of indiscernibles if we want to explicitly mention the ordering associated with the set.

scholarly journals Convexity of sets in metric Abelian groups

2020 ◽  
Vol 32 (6) ◽  
pp. 1477-1486 ◽  
Author(s):  
Włodzimierz Fechner ◽  
Zsolt Páles
Keyword(s):  
Abelian Group ◽  
Structural Properties ◽  
Spectral Radius ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Invariant Metric ◽  
Translation Invariant ◽  
Basic Properties ◽  
The Family ◽  
Cancellation Theorem

AbstractIn the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups, equipped with a translation invariant metric, we define the boundedness, the norm, the modulus of injectivity and the spectral radius of endomorphisms. Beyond the investigation of their properties, our first main goal is an extension of the celebrated Rådström cancellation theorem. Another result generalizes the Neumann invertibility theorem. Next we define the convexity of sets with respect to a family of endomorphisms, and we describe the set-theoretical and algebraic structure of the class of such sets. Given a subset, we also consider the family of endomorphisms that make this subset convex, and we establish the basic properties of this family. Our first main result establishes conditions which imply midpoint convexity. The next main result, using our extension of the Rådström cancellation theorem, presents further structural properties of the family of endomorphisms that make a subset convex.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

scholarly journals A lattice-theoretic characterization of pure subgroups of Abelian groups

2021 ◽  
Author(s):  
M. Ferrara ◽  
M. Trombetti
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Subgroup Lattice ◽  
General Definition ◽  
Short Paper ◽  
Theoretic Characterization ◽  
Definition Of

AbstractLet G be an abelian group. The aim of this short paper is to describe a way to identify pure subgroups H of G by looking only at how the subgroup lattice $$\mathcal {L}(H)$$ L ( H ) embeds in $$\mathcal {L}(G)$$ L ( G ) . It is worth noticing that all results are carried out in a local nilpotent context for a general definition of purity.

On the square subgroups of decomposable torsion-free abelian groups of rank three

2016 ◽  
Vol 7 (4) ◽  
Author(s):  
Fysal Hasani ◽  
Fatemeh Karimi ◽  
Alireza Najafizadeh ◽  
Yousef Sadeghi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractThe square subgroup of an abelian group

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