The Erdős–Ginzburg–Ziv theorem for dihedral groups

Abstract

Let n ≥ 23 be an integer and let D 2n be the dihedral group of order 2n. It is proved that, if g 1 , g 2 , . . . , g 3n is a sequence of 3n elements in D 2n , then there exist 2n distinct indices i 1 , i 2 , . . . , i 2n such that g i 1 g i 2 · · · g i 2n = 1. This result is a sharpening of the famous Erdős-Ginzburg-Ziv theorem for G = D 2n .

Improving the Erdős–Ginzburg–Ziv theorem for some non-abelian groups

“…This was verified for prime n 4001 by Zhuang and Gao [11] and subsequently for all n 23 by Gao and Lu [5]. In Theorem 8 we complete the proof for all n by improving the argument from the latter paper.…”

“…Therefore, all elements y m i −m j are in the subgroup of H of order gcd(u, n). But Up to this point our proof has been the same as that by Gao and Lu [5], but from now on it differs. has |T | = 2n and π(T ) = 1 similarly to sub-subcase 1B(i).…”

“…Following the notation of Gao and Lu [5], let H be the cyclic subgroup of D 2n generated by y, and let N = G \ H . Note that the product of any two elements of N is in H and that every element in N has order 2.…”

“…(See [5,Lemma 7].) Let G be a finite abelian group of order n, r 2 an integer, and S a sequence of n + r − 2 elements in G. If 1 / ∈ n (S), then | n−2 (S)| = | r (S)| r − 1.…”

“…Zhuang and Gao [11] proved the conjecture for dihedral groups of order 2p for p 4001 prime, and Gao and Lu [5] proved it for dihedral groups of order 2n for all n 23. In this paper we extend the proof given by Gao and Lu to prove Conjecture 2 for all dihedral groups, and then use this result to prove the conjecture for dicyclic groups.…”