Let G be a finite abelian group and \(\exp (G)\) the exponent of G. \({\textsf{W}}(G)\) denotes the set of cross numbers of minimal zero-sum sequences over G and \({\textsf{w}}(G)\) denotes the set of all cross numbers of nontrivial zero-sum free sequences over G. It is clear that \({\textsf{W}}(G)\) and \({\textsf{w}}(G)\) are bounded subsets of \(\frac{1}{\exp (G)}\mathbb {N}\) with maximum \({\textsf{K}}(G)\) and \({\textsf{k}}(G)\) , respectively (here \(\textsf{K}(G)\) and \(\textsf{k}(G)\) denote the large and the small cross number of G, respectively). We give results on the structure of \({\textsf{W}}(G)\) and \({\textsf{w}}(G)\) . We first show that both sets contain long arithmetic progressions and that only close to the maximum there might be some gaps. We provide groups for which \({\textsf{W}}(G)\) and \({\textsf{w}}(G)\) actually are arithmetic progressions, and argue that this is rather a rare phenomenon. Finally, we provide some results in case there are gaps.
