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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
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References:

[1] DOI: 10.1016/0022-314X(76)90021-4 · Zbl 0333.05009
[2] Erd?s, Bull. Res. Council 10 (1961)
[3] DOI: 10.1016/S0021-9800(67)80070-X · Zbl 0189.29701
[4] Mann, Addition Theorems (1965)
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