scholarly journals Subset sums in abelian groups

2013 ◽  
Vol 34 (8) ◽  
pp. 1269-1286 ◽  
Author(s):  
Éric Balandraud ◽  
Benjamin Girard ◽  
Simon Griffiths ◽  
Yahya ould Hamidoune
Keyword(s):  
Abelian Groups ◽  
Subset Sums

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].

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 Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

2021 ◽  
Author(s):  
Michele Rossi ◽  
Lea Terracini
Keyword(s):  
Free Group ◽  
Toric Variety ◽  
Class Group ◽  
Abelian Groups ◽  
Toric Varieties ◽  
Picard Group ◽  
Closed Field ◽  
Finitely Generated ◽  
Free Part ◽  
Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

Sign in / Sign up

Export Citation Format

Share Document