On Erdős-Ginzburg-ZIV inverse theorems for dihedral and dicyclic groups

Abstract

Let G be a finite group and exp(G) = lcm{ord(g) ∣ g ∈ G}. A finite unordered sequence of terms from G, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals the identity element of G. We denote by s(G)(or E(G) respectively) the smallest integer l such that every sequence of length at least l has a product-one subsequence of length exp(G)(or ∣G∣ respectively). In this paper, we provide the exact values of s(G) and E(G) for Dihedral and Dicyclic groups and we provide explicit characterizations of all sequences of length s(G) — 1 (or E(G) — 1 respectively) having no product-one subsequence of length exp(G)(or ∣G∣ respectively).