On weighted sequence sums. (English) Zbl 0848.20049
Summary: The main result of this paper has the following consequence. Let \(G\) be an abelian group of order \(n\). Let \(\{x_i:1\leq i\leq 2n-1\}\) be a family of elements of \(G\) and let \(\{w_i:1\leq i\leq n-1\}\) be a family of integers prime relative to \(n\). Then there is a permutation \(\tau\) of \([1,2n-1]\) such that \(\sum_{1\leq i\leq n-1}w_ix_{\tau(i)}=\sum_{1\leq i\leq n-1}w_ix_{\tau(n)}\). Applying this result with \(w_i=1\) for all \(i\), one obtains the Erdös-Ginzburg-Ziv theorem.
MSC:
20K01 | Finite abelian groups |
11B75 | Other combinatorial number theory |
20D60 | Arithmetic and combinatorial problems involving abstract finite groups |
References:
[1] | DOI: 10.1016/0022-314X(76)90021-4 · Zbl 0333.05009 · doi:10.1016/0022-314X(76)90021-4 |
[2] | Erd?s, Bull. Res. Council 10 (1961) |
[3] | DOI: 10.1016/S0021-9800(67)80070-X · Zbl 0189.29701 · doi:10.1016/S0021-9800(67)80070-X |
[4] | Mann, Addition Theorems (1965) |
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