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On cross numbers of minimal zero sequences. (English) Zbl 0866.11018
Let \(G\) be an (additively written) finite abelian group. Let us say that \(S=\{g_1,\dots,g_t\}\) is a minimal zero sequence if \(\sum_{1\leq i\leq t}g_i=0\) and there is not a proper subset \(I\subset \{1,2,\dots,t\}\) for which \(\sum_{i\in I}g_i=0\). The cross number \(k(S)\) is defined by \(k(S)=\sum^t_{i=1} [\text{ord } g_i]^{-1}\). Let \(U(G)=\{S:S\) is a minimal zero sequence} and let \(W(G)=\{k(S): S\in U(G)\}\) and \(W^*(G)= \{k(S):S\in U(G)\) and \(k(S)\leq 1\}\). The authors prove: Let \(G\) be a finite abelian \(p\)-group for some prime \(p\). Suppose that \(G\cong \bigoplus^r_{i=1} C_{n_i}\) with \(1=n_0<n_1\leq \dots\leq n_r=n\). If \(p\) is odd, or \(p=2\) and \(n_{r-1}=n\), then \[ \begin{aligned} W(G)&=\Biggl\{ {\lambda\over n}: 2\leq \lambda\leq nr-\sum^{r-1}_{i=1} {n\over n_i}\Biggr\}.\\ \text{Otherwise} W(G)&= \Biggl\{{\lambda\over n}:2\leq \lambda\leq nr-\sum^{r-1}_{i=1} {n\over{n_i}},\;\lambda\text{ even}\Biggr\}.\end{aligned} \] It is also proved: If \(G\) is a finite abelian group and \(g\) some nonzero element of \(G\), the following conditions are equivalent:
(1) \(G\) is cyclic of prime power order. (2) \(k(S)\leq 1\) for all \(S\in U(G)\) containing \(g\).

11B75 Other combinatorial number theory
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20K01 Finite abelian groups