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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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