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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 \(g\).

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