On Erdős-Ginzburg-Ziv inverse theorems for Dihedral and Dicyclic groups

Abstract

Let $G$ be a finite group and exp$(G)$ = lcm$\{$ord$(g)$$\mid$$g\in 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 $\mathsf{s}(G)$ (or $\mathsf{E}(G)$ respectively) the smallest integer $\ell$ such that every sequence of length at least $\ell$ has a product-one subsequence of length $\exp (G)$ (or $|G|$ respectively). In this paper, we provide the exact values of $\mathsf{s}(G)$ and $\mathsf{E}(G)$ for Dihedral and Dicyclic groups and we provide explicit characterizations of all sequences of length $\mathsf{s}(G)-1$ (or $\mathsf{E}(G)-1$ respectively) having no product-one subsequence of length $\exp (G)$ (or $|G|$ respectively).

To appear in Israel Journal of Mathematics