On the Olson and the Strong Davenport constants

TL;DR

This paper studies the maximal size of zero-sumfree and minimal zero-sum sets in finite abelian groups, providing new bounds and exact values for specific group types, especially p-groups with certain properties.

Contribution

It introduces new bounds for the Olson and Strong Davenport constants for various finite abelian groups, including large-rank p-groups and groups with small exponents.

Findings

Determined Olson constant for several new group classes.

Established new lower bounds for zero-sumfree set sizes.

Calculated the Strong Davenport constant for elementary p-groups of rank at most 2.

Abstract

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.