scholarly journals Zero Sum Partition of Abelian Groups into Sets of the Same Order and its Applications

10.37236/6977 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Sylwia Cichacz
Keyword(s):  
Abelian Group ◽  
Pairwise Disjoint ◽  
Identity Element ◽  
Abelian Groups ◽  
Partition Property ◽  
Zero Sum

We will say that an Abelian group $\Gamma$ of order $n$ has the $m$-zero-sum-partition property ($m$-ZSP-property) if $m$ divides $n$, $m\geq 2$ and there is a partition of $\Gamma$ into pairwise disjoint subsets $A_1, A_2,\ldots , A_t$, such that $|A_i| = m$ and $\sum_{a\in A_i}a = g_0$ for $1 \leq i \leq t$, where $g_0$ is the identity element of $\Gamma$.In this paper we study the $m$-ZSP property of $\Gamma$. We show that $\Gamma$ has the $m$-ZSP property if and only if $m\geq 3$ and $|\Gamma|$ is odd or $\Gamma$ has more than one involution. We will apply the results to the study of group distance magic graphs as well as to generalized Kotzig arrays.

scholarly journals Zero-sum subsets of decomposable sets in Abelian groups

2020 ◽  
Vol 30 (1) ◽  
pp. 15-25
Author(s):  
T. Banakh ◽  
◽  
A. Ravsky ◽  
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Open Problem ◽  
Abelian Groups ◽  
The Other ◽  
Other Hand ◽  
Decomposable Sets ◽  
Empty Subset ◽  
Zero Sum

A subset D of an abelian group is decomposable if ∅≠D⊂D+D. In the paper we give partial answers to an open problem asking whether every finite decomposable subset D of an abelian group contains a non-empty subset Z⊂D with ∑Z=0. For every n∈N we present a decomposable subset D of cardinality |D|=n in the cyclic group of order 2n−1 such that ∑D=0, but ∑T≠0 for any proper non-empty subset T⊂D. On the other hand, we prove that every decomposable subset D⊂R of cardinality |D|≤7 contains a non-empty subset T⊂D of cardinality |Z|≤12|D| with ∑Z=0. For every n∈N we present a subset D⊂Z of cardinality |D|=2n such that ∑Z=0 for some subset Z⊂D of cardinality |Z|=n and ∑T≠0 for any non-empty subset T⊂D of cardinality |T|<n=12|D|. Also we prove that every finite decomposable subset D of an Abelian group contains two non-empty subsets A,B such that ∑A+∑B=0.

scholarly journals Unextendible Sequences in Finite Abelian Groups

10.37236/899 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Jujuan Zhuang
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Minimal Length ◽  
Finite Abelian Groups ◽  
Zero Sum

Let $G=C_{n_1}\oplus \ldots \oplus C_{n_r}$ be a finite abelian group with $r=1$ or $1 < n_1|\ldots|n_r$, and let $S=(a_1,\ldots,a_t)$ be a sequence of elements in $G$. We say $S$ is an unextendible sequence if $S$ is a zero-sum free sequence and for any element $g\in G$, the sequence $Sg$ is not zero-sum free any longer. Let $L(G)=\lceil \log_2{n_1}\rceil+\ldots+\lceil \log_2{n_r}\rceil$ and $d^*(G)=\sum_{i=1}^r(n_i-1)$, in this paper we prove, among other results, that the minimal length of an unextendible sequence in $G$ is not bigger than $L(G)$, and for any integer $k$, where $L(G)\leq k \leq d^*(G)$, there exists at least one unextendible sequence of length $k$.

On subset sums of zero-sum free sets of abelian groups

2019 ◽  
Vol 15 (03) ◽  
pp. 645-654 ◽  
Author(s):  
Jiangtao Peng ◽  
Wanzhen Hui ◽  
Yuanlin Li ◽  
Fang Sun
Keyword(s):  
Abelian Group ◽  
Nonempty Subset ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Subset Sums ◽  
Zero Sum ◽  
Free Sets

Let [Formula: see text] be a finite abelian group and [Formula: see text] be a subset of [Formula: see text]. Let [Formula: see text] denote the set of group elements which can be expressed as a sum of a nonempty subset of [Formula: see text]. We say that [Formula: see text] is zero-sum free if [Formula: see text]. In this paper, we show that if [Formula: see text] is zero-sum free with [Formula: see text], then [Formula: see text]. Furthermore, if [Formula: see text] is zero-sum free and [Formula: see text], with the assumption that the subgroup generated by [Formula: see text] is noncyclic and the order [Formula: see text], we are able to show that [Formula: see text].

scholarly journals On Short Zero-Sum Subsequences of Zero-Sum Sequences

10.37236/2602 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Yushuang Fan ◽  
Weidong Gao ◽  
Guoqing Wang ◽  
Qinghai Zhong ◽  
Jujuan Zhuang
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Finite Abelian Groups ◽  
Special Groups ◽  
Large Exponent ◽  
Zero Sum

Let $G$ be a finite abelian group of exponent $\exp(G)$. By $D(G)$ we denote the smallest integer $d\in \mathbb N$ such that every sequence over $G$ of length at least $d$ contains a nonempty zero-sum subsequence. By $\eta(G)$ we denote the smallest integer $d\in \mathbb N$ such that every sequence over $G$ of length at least $d$ contains a zero-sum subsequence $T$ with length $|T|\in [1,\exp(G)]$, such a sequence $T$ will be called a short zero-sum sequence. Let $C_0(G)$ denote the set consists of all integer $t\in [D(G)+1,\eta(G)-1]$ such that every zero-sum sequence of length exactly $t$ contains a short zero-sum subsequence. In this paper, we investigate the question whether $C_0(G)\neq \emptyset$ for all non-cyclic finite abelian groups $G$. Previous results showed that $C_0(G)\neq \emptyset$ for the groups $C_n^2$ ($n\geq 3$) and $C_3^3$. We show that more groups including the groups $C_m\oplus C_n$ with $3\leq m\mid n$, $C_{3^a5^b}^3$, $C_{3\times 2^a}^3$, $C_{3^a}^4$ and $C_{2^b}^r$ ($b\geq 2$) have this property. We also determine $C_0(G)$ completely  for some groups including the groups of rank two, and some special groups with large exponent.

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

Representation of zero-sum invariants by sets of zero-sum sequences over a finite abelian group

2021 ◽  
Author(s):  
Weidong Gao ◽  
Siao Hong ◽  
Wanzhen Hui ◽  
Xue Li ◽  
Qiuyu Yin ◽  
...  
Keyword(s):  
Abelian Group ◽  
Finite Abelian Group ◽  
Zero Sum

scholarly journals ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

2011 ◽  
Vol 10 (03) ◽  
pp. 377-389
Author(s):  
CARLA PETRORO ◽  
MARKUS SCHMIDMEIER
Keyword(s):  
Abelian Group ◽  
Prime Number ◽  
Maximal Ideal ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Uniserial Ring

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

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