On minimal product-one sequences of maximal length over Dihedral and Dicyclic groups
Commun. Korean Math. Soc. 2020 Vol. 35, No. 1, 83-116
Published online January 31, 2020
Jun Seok Oh, Qinghai Zhong
University of Graz; 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 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
MSC numbers : 20D60, 20M13, 11B75, 11P70
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