Inverse zero-sum problems in finite Abelian p-groups

TL;DR

This paper investigates the minimal number of maximal order elements in zero-sumfree sequences within finite Abelian p-groups, introducing a new parameter and establishing bounds that improve existing results on sequence structure.

Contribution

It introduces a new number for analyzing zero-sumfree sequences and provides bounds that enhance understanding of the sequence composition in finite Abelian p-groups.

Findings

A new parameter for zero-sumfree sequences is introduced.

Bounds for the number of maximal order elements are established.

Improves previous results on the minimal number of such elements.

Abstract

In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, the method that we use here enables us to show that, if we denote by exp(G) the exponent of the finite Abelian p-group G which is considered, then a zero-sumfree sequence S with maximal possible length in G must contain at least exp(G)-1 elements of maximal order, which improves a previous result of W. Gao and A. Geroldinger.