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combinatorics

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

arXiv:1904.13171

summary

The paper determines the exact values of the Erdős‑Ginzburg‑Ziv constants s(G) and E(G) for dihedral and dicyclic groups and characterizes the extremal sequences that avoid the required product‑one subsequences.

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

Topics & keywords

#erdos-ginzburg-zhiv theorem#dihedral groups#dicyclic groups#zero-sum sequences#product-one sequences#inverse problemss(G)E(G)group exponentproduct-one subsequenceextremal sequencesfinite non‑abelian groups