  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.
