On Cross Numbers of Minimal Zero Sequences in Certain Cyclic Groups

Abstract

Let $p$ and $q$ be distinct primes with $p>q$ and $n$ a positive integer. In this paper, we consider the set of possible cross numbers for the cyclic groups ${\mathbb{Z}}_{2p^n}$ and ${\mathbb{Z}}_{pq}$. We completely determine this set for ${\mathbb{Z}}_{2p^n}$ and also ${\mathbb{Z}}_{pq}$ for $q=3,q=5$ and the case where $p$ is sufficiently larger than $g$. We view the latter result in terms of an upper bound for this set developed in a paper of Geroldinger and Schneider [8] and show precisely when this upper bound is an equality.