The Erdős-Ginzburg-Ziv theorem for finite solvable groups.
The Davenport constant \(D(G)\) of a group \(G\) is the smallest integer \(d\) such that every sequence of \(d\) elements (repetition allowed) in \(G\) contains a nonempty subsequence (permutation allowed) of product \(=1\). Denote by \(s(G)\) the smallest integer \(t\) such that every sequence of \(t\) elements in \(G\) contains a subsequence of length \(n\) and product \(=1\). In 1961, Erdős, Ginzburg, Ziv showed that \(s(G)\leq 2n-1\) for every finite solvable group \(G\). In this paper the authors substantially improve this, by proving the following Theorem. If \(G\) is a non-cyclic solvable group of order \(n\), then \(s(G)\leq\tfrac{7}{4}n-1\). Moreover, they conjecture \(\tfrac{3}{2}n\) as the best upper bound for \(s(G)\). Reviewer's remark: For a finite Abelian group of order \(n\), the first author proved that \(s(G)=n-1+D(G)\). Another conjecture is that this equality holds for every finite group \(G\) of order \(n\).
scientific article; zbMATH DE number 5673211
scientific article