## The cross number of finite Abelian groups.(English)Zbl 0814.20033

Let $$G$$ be a finite abelian group, $$G = C_{n_ 1} \oplus \cdots \oplus C_{n_ r}$$ its direct decomposition into cyclic groups of prime power order, and $$\exp(G)$$ the exponent of $$G$$. Let $${\mathcal U}(G)$$ denote the set of sequences $$S$$ in $$G$$ with sum zero, for which no proper subsequence has sum zero. For a sequence $$S = (g_ 1,\dots,g_ m)$$ of elements in $$G$$ we define its cross number $$k(S)$$ by $$k(S) = \sum^ m_{i = 1} {1\over \text{ord}(g_ i)}$$ and we set $$K(G) = \exp(G)\max\{k(S)\mid S\in {\mathcal U}(G)\}$$ and $$K^*(G) = 1+ \exp(G) \sum^ r_{i = 1} {n_ i - 1\over n_ i}$$. $$K(G)$$ is called the cross number of $$G$$. It is easy to see that $$K(G) \geq K^*(G)$$. U. Krause and C. Zahlten conjectured that equality holds for all cyclic groups. In this paper, the author concentrates on $$p$$-groups and obtains the following theorem: $$K(G) = K^*(G)$$ for any finite abelian $$p$$-group $$G$$. The general case remains open. In particular no finite abelian group $$G$$ with $$K(G) > K^*(G)$$ is known.
In order to prove the theorem, the author introduces a variant of the cross number which is easier to handle. Let $$G = \bigoplus^ r_{i = 1} C_{n_ i}$$ be a finite abelian group with prime powers $$n_ 1,\dots,n_ r$$ and let $$u(G)$$ denote the set of sequences $$S$$ in $$G$$, for which no subsequence $$S' \subseteq S$$ has sum zero. Set $$k(G) = \max\{k(S)\mid S \in u(G)\}$$ and $$k^*(G) = \sum^ r_{i = 1} {n_ i - 1\over n_ i}$$. The proof of the theorem is reduced to showing that if $$G$$ is a finite abelian $$p$$-group, then $$k(G) = k^*(G)$$.

### MSC:

 20K01 Finite abelian groups 20D60 Arithmetic and combinatorial problems involving abstract finite groups
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