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.