For a finite Abelian group \(G\), the following definitions are used: any finite sequence \(S=(g_1,\dots,g_l)\) is a sequence in \(G\) of length \(l\). For such a sequence, \(s\in G\) is a ‘subsum’ of \(S\) when \(s\in\{\sum_{i\in I}g_i:\emptyset\varsubsetneq I\subseteq\{1,\dots,l\}\}\). If \(0\) is not a subsum of \(S\), then \(S\) is a ‘zero-sumfree sequence’. The ‘cross number of a sequence’ \(S\) in \(G\) is \(k(S)=\sum_{i=1}^l\tfrac{1}{\text{ord}(g_i)}\), and, the ‘little cross number’ \(k(G)\) of \(G\) is \(k(G)=\max\{k(S):S\) zero-sumfree sequence in \(G\}\).
The main result reads as follows: Theorem 2.1. Let \(G\simeq C_{n_1}\oplus\cdots\oplus C_{n_r}\), with \(1<n_1\mid\cdots\mid n_r\in\mathbb{N}\), be a finite Abelian group with \(\exp(G)=n\) and \(\tau(G)=m\). For every zero-sumfree sequence \(S\) in \(G\) reaching the maximum \(k(S)=k(G)\), and being of minimal length regarding this property, the \(m\)-tuple \(x=(|S_{d_1}|,\dots,|S_{d_m}|)\) is an element of the polytope \(\mathbb{P}_G\cap\mathbb{H}_G\) where \(\mathbb{P}_G=\{x\in\mathbb{N}^m:f_d(x)\geq 0,\;g_d(x)\geq 0,\;d\in\mathcal D_n\}\) and \(\mathbb{H}_G=\{x\in\mathbb{N}^m:h(x)\geq 0\}\). – Here (among others) \(\tau(G)\) denotes the number of positive divisors of \(n\).
Corollary 2.2. For every finite Abelian group \(G\), one has the following upper bound: \[ k(G)\leq\max_{x\in\mathbb{P}_G}\left(\sum_{i=1}^m\frac{x_{d_i}}{d_i}\right). \]