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Representation of finite abelian group elements by subsequence sums

David J. GrynkiewiczLuz E. MarchanOscar Ordaz — 2009

Journal de Théorie des Nombres de Bordeaux

Let G C n 1 ... C n r be a finite and nontrivial abelian group with n 1 | n 2 | ... | n r . A conjecture of Hamidoune says that if W = w 1 · ... · w n is a sequence of integers, all but at most one relatively prime to | G | , and S is a sequence over G with | S | | W | + | G | - 1 | G | + 1 , the maximum multiplicity of S at most | W | , and σ ( W ) 0 mod | G | , then there exists a nontrivial subgroup H such that every element g H can be represented as a weighted subsequence sum of the form g = n i = 1 w i s i , with s 1 · ... · s n a subsequence of S . We give two examples showing this does not hold in general, and characterize the counterexamples...

On the Olson and the Strong Davenport constants

Oscar OrdazAndreas PhilippIrene SantosWolfgang A. Schmid — 2011

Journal de Théorie des Nombres de Bordeaux

A subset S of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of S is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, p -groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general...

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