On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups

Commun. Korean Math. Soc. Published online August 27, 2019

Jun Seok Oh and Qinghai Zhong
University of Graz

Abstract : Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where the repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals the identity element of $G$. The large Davenport constant $\mathsf D (G)$ is the maximal length of a minimal product-one sequence, that is a product-one sequence which cannot be factored into two non-trivial product-one subsequences. We provide explicit characterizations of all minimal product-one sequences of length $\mathsf D (G)$ over Dihedral and Dicyclic groups. Based on these characterizations we study the unions of sets of lengths of the monoid of product-one sequences over these groups.

Keywords : product-one sequences, Davenport constant, Dihedral groups, Dicyclic groups, sets of lengths, unions of sets of lengths