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arXiv · 1904.13171

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

Jun Seok Oh·Qinghai Zhong
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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).

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Jun Seok Oh, Qinghai Zhong. 2019-04-30. On Erd\H{o}s-Ginzburg-Ziv inverse theorems for Dihedral and Dicyclic groups. https://arxiv.org/abs/1904.13171

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