A note on Bass' conjecture

Abstract

For a finite group $G$, we denote by $\mathsf{d}(G)$ and by $\mathsf{E}(G)$, respectively, the small Davenport constant and the Gao constant of $G$. Let $C_n$ be the cyclic group of order $n$ and let $G_{m,n,s}=C_n{⋊}_s{C}_m$ be a metacyclic group. In [J. Bass; {\em Improving the Erdős-Ginzburg-Ziv theorem for some non-abelian groups.} J. Number Theory {\bf 126} (2007), 217-236, Conjecture 17], Bass conjectured that $\mathsf{d}(G_{m,n,s})=m+n-2$ and $\mathsf{E}(G_{m,n,s})=mn+m+n-2$ provided $or{d}_n(s)=m$. In this paper, we show that the assumption $or{d}_n(s)=m$ is essential and cannot be removed. Moreover, if we suppose that Bass' conjecture holds for $G_{m,n,s}$ and the $mn$-product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for $G_{2m,2n,r}$, where $r^2\equiv s(\mathrm{mod}n)$.