Combinatorial Approaches to Minimal Zero Sequences of Finite Abelian Groups, and a Surprising Connection

This paper presents a fresh new combinatorial approach to the study of minimal zero sequences of a finite abelian group. We begin by presenting some general algebraic theory aimed at reducing the search space of potential minimal zero sequences for a certain class of groups, as well as describe the optimal algorithm for generating all minimal zero sequences of such a group. We continue by giving a closed form solution for the number of atoms in an infinite family of groups. Then, by introducing the concept of the dimension class of an atom in the block monoid of a group, we are able to give a partial description of another infinite family of groups. Finally, we conclude by making a strong connection between the study of minimal zero sequences and the polynomials of a group which are invariant under a certain class of cyclic variable substitutions.