Extremal product-one free sequences in Cq⋊sCm

Abstract

Let G be a finite group. The Davenport constant of G is the smallest positive integer d such that every sequence over G with d elements has a non-empty subsequence with product 1. Let Cn be the cyclic group of order n. Bass [1] showed that the Davenport constant of the metacyclic group Cq⋊sCm, where q is a prime number and ordq(s)=m≥2, is m+q−1. In this paper, we determine the form of all sequences S of Cq⋊sCm, with q+m−2 elements that are product-one free.