Abstract
Let G be a finite group, written multiplicatively. Define \({\mathsf{E}(G)}\) to be the minimal integer t such that every sequence of t elements (repetition allowed) in G contains a subsequence with length \({|G|}\) and with product one (in some order). Let p be the smallest prime divisor of \({|G|}\). In this paper we prove that if G is a noncyclic nilpotent group then \({\mathsf{E}(G) \le |G|+\frac{|G|}{p}+p-2}\), which confirms partially a conjecture by Gao and Li. We also determine the exact value of \({\mathsf{E}(G)}\) for \({G=C_{p}\ltimes C_{pn}}\) when p is a prime, which confirms partially another conjecture by Zhuang and Gao.
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Han, D. The Erdős–Ginzburg–Ziv theorem for finite nilpotent groups. Arch. Math. 104, 325–332 (2015). https://doi.org/10.1007/s00013-015-0739-4
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DOI: https://doi.org/10.1007/s00013-015-0739-4