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

Let G be a finite group of order n , and let S = ( a 1 , … , a k ) be a sequence of elements in G . We call S a 1-product sequence if 1 = ∏ i = 1 k a τ ( 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 2 p then s ( G ) = | G | + D ( G ) − 1 = 3 p , where p ≥ 4001 is a prime.