# zbMATH — the first resource for mathematics

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
##### Keywords:
finite abelian group; minimal zero sequence; cross number