Erdős–Ginzburg–Ziv theorem for dihedral groups of large prime index

Abstract

Let G be a finite group of order n, and let S=(a1,…,ak) be a sequence of elements in G. We call S a 1-product sequence if 1=∏i=1kaτ(i) holds for some permutation τ of {1,…,k}. By s(G) we denote the smallest integer t such that, every sequence of t elements in G contains a 1-product subsequence of length n. By D(G) we denote the smallest integer d such that every sequence of d elements in G contains a nonempty 1-product subsequence. We prove that if G is a non-Abelian group of order 2p then s(G)=|G|+D(G)−1=3p, where p≥4001 is a prime.