THE MAXIMUM SIZE OF -SUM-FREE SETS IN CYCLIC GROUPS

2018 ◽  
Vol 99 (2) ◽  
pp. 184-194
Author(s):  
BÉLA BAJNOK ◽  
RYAN MATZKE
Keyword(s):  
Number Theory ◽  
Abelian Group ◽  
Explicit Formula ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Maximum Size ◽  
Critical Pair ◽  
Cyclic Groups ◽  
Free Subset ◽  
Free Sets

A subset$A$of a finite abelian group$G$is called$(k,l)$-sum-free if the sum of$k$(not necessarily distinct) elements of$A$never equals the sum of$l$(not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a$(k,l)$-sum-free subset in$G$for all$k$and$l$in the case when$G$is cyclic by proving that it suffices to consider$(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’,Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a$(k,l)$-sum-free subset of an abelian group’,Int. J. Number Theory 5(6) (2009), 953–971].

scholarly journals ON THE MAXIMUM SIZE OF A (k,l)-SUM-FREE SUBSET OF AN ABELIAN GROUP

2009 ◽  
Vol 05 (06) ◽  
pp. 953-971 ◽  
Author(s):  
BÉLA BAJNOK
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Maximum Size ◽  
Finite Abelian Groups ◽  
Free Subset ◽  
Free Set ◽  
Free Sets

A subset A of a given finite abelian group G is called (k,l)-sum-free if the sum of k (not necessarily distinct) elements of A does not equal the sum of l (not necessarily distinct) elements of A. We are interested in finding the maximum size λk,l(G) of a (k,l)-sum-free subset in G. A (2,1)-sum-free set is simply called a sum-free set. The maximum size of a sum-free set in the cyclic group ℤn was found almost 40 years ago by Diamanda and Yap; the general case for arbitrary finite abelian groups was recently settled by Green and Ruzsa. Here we find the value of λ3,1(ℤn). More generally, a recent paper by Hamidoune and Plagne examines (k,l)-sum-free sets in G when k - l and the order of G are relatively prime; we extend their results to see what happens without this assumption.

scholarly journals The Number of Subgroup Chains of Finite Nilpotent Groups

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1537 ◽  
Author(s):  
Lingling Han ◽  
Xiuyun Guo
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Recursive Formula ◽  
Nilpotent Groups ◽  
Classification Problem ◽  
Finite Abelian Groups ◽  
Counting Problem ◽  
Cyclic Groups ◽  
Fuzzy Subgroups

In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

scholarly journals The Sharp Threshold for Maximum-Size Sum-Free Subsets in Even-Order Abelian Groups

2015 ◽  
Vol 24 (4) ◽  
pp. 609-640 ◽  
Author(s):  
NEAL BUSHAW ◽  
MAURÍCIO COLLARES NETO ◽  
ROBERT MORRIS ◽  
PAUL SMITH
Keyword(s):  
Recent Work ◽  
Abelian Groups ◽  
Maximum Size ◽  
Constant Factor ◽  
Sharp Threshold ◽  
Even Order ◽  
Free Subset ◽  
Free Sets

We study sum-free sets in sparse random subsets of even-order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the caseG= ℤ2n, and who obtained a weaker threshold (up to a constant factor) in general.

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 The number of additive triples in subsets of abelian groups

2016 ◽  
Vol 160 (3) ◽  
pp. 495-512 ◽  
Author(s):  
WOJCIECH SAMOTIJ ◽  
BENNY SUDAKOV
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
General Question ◽  
Free Subset

AbstractA set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elementsx,y,zwithx+y=z. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a setAof elements of an abelian groupGhas cardinalitya. How many Schur triples mustAcontain? Moreover, which sets ofaelements ofGhave the smallest number of Schur triples? In this paper, we answer these questions for various groupsGand ranges ofa.

scholarly journals On finite loops whose inner mapping groups are Abelian II

2005 ◽  
Vol 71 (3) ◽  
pp. 487-492
Author(s):  
Markku Niemenmaa
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Prime Power ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Prime Power Order ◽  
Finite Abelian Groups ◽  
Cyclic Groups ◽  
Inner Mapping Group ◽  
Mapping Group

If the inner mapping group of a loop is a finite Abelian group, then the loop is centrally nilpotent. We first investigate the structure of those finite Abelian groups which are not isomorphic to inner mapping groups of loops and after this we show that if the inner mapping group of a loop is isomorphic to the direct product of two cyclic groups of the same odd prime power order pn, then our loop is centrally nilpotent of class at most n + 1.

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.

Stable cohomotopy and cobordism of abelian groups

1981 ◽  
Vol 90 (2) ◽  
pp. 273-278 ◽  
Author(s):  
C. T. Stretch
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Formal Group ◽  
Characteristic Class ◽  
Burnside Ring ◽  
Classifying Space ◽  
Formal Group Law ◽  
Stable Cohomotopy ◽  
Group Law

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

Criteria for Total Projectivity

1981 ◽  
Vol 33 (4) ◽  
pp. 817-825 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Abelian Groups ◽  
General Criterion ◽  
Direct Sums ◽  
Primary Abelian Group ◽  
Cyclic Groups ◽  
Direct Sum ◽  
Totally Projective Groups ◽  
Torsion Groups

All groups herein are assumed to be abelian. It was not until the 1940's that it was known that a subgroup of an infinite direct sum of finite cyclic groups is again a direct sum of cyclics. This result rests on a general criterion due to Kulikov [7] for a primary abelian group to be a direct sum of cyclic groups. If G is p-primary, Kulikov's criterion presupposes that G has no elements (other than zero) having infinite p-height. For such a group G, the criterion is simply that G be the union of an ascending sequence of subgroups Hn where the heights of the elements of Hn computed in G are bounded by some positive integer λ(n). The theory of abelian groups has now developed to the point that totally projective groups currently play much the same role, at least in the theory of torsion groups, that direct sums of cyclic groups and countable groups played in combination prior to the discovery of totally projective groups and their structure beginning with a paper by R. Nunke [11] in 1967.

On Nilpotent Products of Cyclic Groups

1960 ◽  
Vol 12 ◽  
pp. 447-462 ◽  
Author(s):  
Ruth Rebekka Struik
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Finite Abelian Group ◽  
Nilpotent Groups ◽  
Multiplication Table ◽  
Direct Products ◽  
Free Products ◽  
Cyclic Groups ◽  
Special Cases ◽  
Product Of Groups

In this paper G = F/Fn is studied for F a free product of a finite number of cyclic groups, and Fn the normal subgroup generated by commutators of weight n. The case of n = 4 is completely treated (F/F2 is well known; F/F3 is completely treated in (2)); special cases of n > 4 are studied; a partial conjecture is offered in regard to the unsolved cases. For n = 4 a multiplication table and other properties are given.The problem arose from Golovin's work on nilpotent products ((1), (2), (3)) which are of interest because they are generalizations of the free and direct product of groups: all nilpotent groups are factor groups of nilpotent products in the same sense that all groups are factor groups of free products, and all Abelian groups are factor groups of direct products. In particular (as is well known) every finite Abelian group is a direct product of cyclic groups. Hence it becomes of interest to investigate nilpotent products of finite cyclic groups.

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