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 D2n be the Dihedral Group of order 2n and Q4n be the Dicyclic Group of order 4n. Zhuang and Gao (2005) showed that D(D2n)=n+1 and Bass (2007) showed that D(Q4n)=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 D2n or Q4n for some n.