On the existence of zero-sum subsequences of distinct lengths

TL;DR

This paper characterizes short normal sequences over finite Abelian p-groups, confirming Gao's conjecture in several cases by leveraging combinatorial and algebraic methods, including Alon's Nullstellensatz.

Contribution

It provides a positive resolution of Gao's conjecture for various finite Abelian p-groups using novel applications of combinatorial theorems and algebraic techniques.

Findings

Confirmed Gao's conjecture for elementary p-groups.

Extended the conjecture's validity to rank two finite Abelian groups.

Connected the problem to properties of regular subgraphs and algebraic nullstellensatz.

Abstract

In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups. Our main result is deduced from a theorem of Alon, Friedland and Kalai, originally proved so as to study the existence of regular subgraphs in almost regular graphs. In the special case of elementary p-groups, Gao's conjecture is solved using Alon's Combinatorial Nullstellensatz. To conclude, we show that, assuming every integer satisfies Property B, this conjecture holds in the case of finite Abelian groups of rank two.