On Erd\H{o}s-Ginzburg-Ziv inverse theorems for Dihedral and Dicyclic groups

TL;DR

This paper determines the exact values of the Erdős-Ginzburg-Ziv constants for Dihedral and Dicyclic groups, providing explicit characterizations of sequences lacking certain product-one subsequences.

Contribution

It offers the first precise calculations of these constants for specific non-abelian groups and characterizes sequences that do not contain particular product-one subsequences.

Findings

Exact values of (G) and (G) for Dihedral and Dicyclic groups

Characterizations of sequences without product-one subsequences of specific lengths

Extension of inverse theorems to non-abelian 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).