Extremal product-one free sequences in Dihedral and Dicyclic groups

Abstract

Let $G$ be a finite group, written multiplicatively. The Davenport constant of $G$ is the smallest positive integer $D(G)$ such that every sequence of $G$ with $D(G)$ elements has a non-empty subsequence with product $1$. Let $D_{2n}$ be the Dihedral Group of order $2n$ and $Q_{4n}$ be the Dicyclic Group of order $4n$. J. J. Zhuang and W. Gao (European J. Combin. 26 (2005), 1053-1059) showed that $D(D_{2n}) = n+1$ and J. Bass (J. Number Theory 126 (2007), 217-236) showed that $D(Q_{4n}) = 2n+1$. In this paper, we give explicit characterizations of all sequences $S$ of $G$ such that $|S| = D(G) - 1$ and $S$ is free of subsequences whose product is $1$, where $G$ is equal to $D_{2n}$ or $Q_{4n}$ for some $n$.