On the Olson and the Strong Davenport constants

Ordaz, Oscar; Philipp, Andreas; Santos, Irene; Schmid, Wolfgang A.

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 3, pp. 715-750.

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 results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary $p$-groups of rank at most $2$, paralleling and building on recent results on this problem for the Olson constant.

Soit $S$ un sous-ensemble d’un groupe abélien fini, noté additivement. Si $0$ n’est pas une sous-somme (non vide) de $S$, on dit que $S$ est un ensemble sans sous-somme nulle. Nous examinons la cardinalité maximale d’un ensemble sans sous-somme nulle, c’est-à-dire la (petite) constante d’Olson. Nous déterminons la cardinalité maximale d’un tel ensemble pour plusieurs types de groupes ; en particulier, les $p$-groupes dont le rang est suffisament grand relativement à l’exposant et plus particulièrement tous les groupes dont l’exposant est au plus $5$. Nous obtenons ces résultats comme des cas particuliers de résultats plus généraux, donnant des bornes inférieures pour la cardinalité d’un ensemble sans sous-somme nulle pour des groupes variés. Nous examinons la qualité de ces bornes en considérant des cas explicites, avec l’aide d’un ordinateur. De plus, nous examinons une notion très proche de la constante d’Olson : la cardinalité maximale d’un ensemble minimal de somme nulle, c’est-à-dire la constante de Davenport forte. En particulier, nous déterminons la valeur de cette constante pour les $p$-groupes élémentaires dont le rang est au plus $2$, en utilisant des résultats récents sur la constante d’Olson.

Classification: 11B30, 11B50, 20K01
Keywords: Davenport constant, Strong Davenport constant, Olson constant, zero-sumfree, zero-sum problem

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