Group Partitions and Mixed Perfect Codes

1975 ◽  
Vol 18 (1) ◽  
pp. 57-60 ◽  
Author(s):  
Bernt Lindström
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Perfect Codes ◽  
Necessary And Sufficient

AbstractLet G be a finite abelian group of the order pr and type (p, …, p), where p is a prime. A necessary and sufficient condition is determined for the existence of subgroups G1, G2, ⋯, Gn, one of the order pa and the rest of the order pb, such that G = G1 ∪ G2 ∪ ⋯ ∪ Gn and Gi, ∩ Gj,= {θ} when i ≠ j.

On the Structure of G(H, k)

2014 ◽  
Vol 21 (02) ◽  
pp. 317-330 ◽  
Author(s):  
Guixin Deng ◽  
Pingzhi Yuan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Explicit Formula ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Directed Edge ◽  
Necessary And Sufficient

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd (p, k)=1.

scholarly journals Commutative feebly nil-clean group rings

2019 ◽  
Vol 11 (2) ◽  
pp. 264-270
Author(s):  
Peter V. Danchev
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Group Ring ◽  
Group Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Clean Rings ◽  
Unital Ring ◽  
Necessary And Sufficient

Abstract An arbitrary unital ring R is called feebly nil-clean if any its element is of the form q + e − f, where q is a nilpotent and e, f are idempotents with ef = fe. For any commutative ring R and any abelian group G, we find a necessary and sufficient condition when the group ring R(G) is feebly nil-clean only in terms of R, G and their sections. Our result refines establishments due to McGovern et al. in J. Algebra Appl. (2015) on nil-clean rings and Danchev-McGovern in J. Algebra (2015) on weakly nil-clean rings, respectively.

Edwards-Walsh resolutions of complexes and Abelian groups

2000 ◽  
Vol 62 (3) ◽  
pp. 407-416 ◽  
Author(s):  
Katsuya Yokoi
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Simplicial Complexes ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

We give a necessary and sufficient condition for the existence of an Edwards-Walsh resolution of a complex. Our theorem is an extension of Dydak-Walsh's theorem to all simplicial complexes of dimension ≥ n + 2. We also determine the structure of an Abelian group with the Edwards-Walsh condition, (which was introduced by Koyama and the author).

scholarly journals On the Equationally Artinian Groups

2020 ◽  
pp. 583-595
Author(s):  
Mohammad Shahryari ◽  
Javad Tayyebi
Keyword(s):  
Abelian Group ◽  
Normal Subgroup ◽  
Large Class ◽  
Finite Extension ◽  
Finite Union ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Closed Sets ◽  
Necessary And Sufficient

In this article, we study the property of being equationally Artinian in groups. We define the radical topology corresponding to such groups and investigate the structure of irreducible closed sets of these topologies. We prove that a finite extension of an equationally Artinian group is again equationally Artinian. We also show that a quotient of an equationally Artinian group of the form G[t] by a normal subgroup which is a finite union of radicals, is again equationally Artnian. A necessary and sufficient condition for an Abelian group to be equationally Artinian will be given as the last result. This will provide a large class of examples of equationally Artinian groups

Perfect codes in poset block spaces

2017 ◽  
Vol 10 (01) ◽  
pp. 1750005
Author(s):  
B. K. Dass ◽  
Namita Sharma ◽  
Rashmi Verma
Keyword(s):  
Block Structure ◽  
Block Code ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Perfect Codes ◽  
Minimal Elements ◽  
Block Spaces ◽  
Hamming Metric ◽  
Limited Class ◽  
Necessary And Sufficient

There is a limited class of perfect codes with respect to the classical Hamming metric. There are other kind of metrics with respect to which perfect codes have been investigated viz. poset metric, block metric and poset block metric. Given the minimal elements of a poset, a necessary and sufficient condition for [Formula: see text]-perfectness of a poset block code has been derived. A necessary and sufficient condition for a poset block code to be [Formula: see text]-perfect has also been considered. Further, for each [Formula: see text], [Formula: see text], a sufficient condition that ensures the existence of a poset block structure which turns a given code into an [Formula: see text]-perfect poset block code has been obtained. Several illustrations of well known codes to be [Formula: see text]-perfect for specific values of [Formula: see text] have been explored.

scholarly journals On orderability of topological groups

1985 ◽  
Vol 8 (4) ◽  
pp. 747-754
Author(s):  
G. Rangan
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Group Structure ◽  
Locally Compact Abelian Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Topological Groups ◽  
Locally Compact ◽  
Necessary And Sufficient ◽  
Totally Disconnected

A necessary and sufficient condition for a topological group whose topology can be induced by a total order compatible with the group structure is given and such groups are called ordered or orderable topological groups. A separable totally disconnected ordered topological group is proved to be non-archimedean metrizable while the converse is shown to be false by means of an example. A necessary and sufficient condition for a no-totally disconnected locally compact abelian group to be orderable is also given.

Isometry Groups with Surface-Orthogonal Trajectories

1967 ◽  
Vol 22 (9) ◽  
pp. 1351-1355 ◽  
Author(s):  
Bernd Schmidt
Keyword(s):  
Abelian Group ◽  
Tangent Space ◽  
Isometry Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Isometry Groups ◽  
Linearly Independent ◽  
Necessary And Sufficient

It is shown that the trajectories of an isometry group admit orthogonal surfaces if the sub-group of stability leaves no vector in the tangent space of the trajectories fixed. A necessary and sufficient condition is given that the trajectories of an Abelian group admit orthogonal surfaces.In spacetimes which admit an Abelian G2 of isometries, the trajectories admit orthogonal 2-surfaces if a timelike congruence exists with the following properties: the curves lie in the trajectories and are invariant under G2; ωα and üα are linearly independent and orthogonal to the trajectories.

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

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