For a finite group $G$, we denote by ${ \sf d}(G)$ and by ${ \sf E}(G)$, respectively, the small Davenport constant and the Gao constant of $G$. Let $C_n$ be the cyclic group of order $n$ and let $G_{ m,n,s} = C_n \rtimes_s C_m$ be a metacyclic group. In [J. Bass; { \em Improving the Erd\H{ o}s-Ginzburg-Ziv theorem for some non-abelian groups.} J. Number Theory { \bf 126} (2007), 217-236, Conjecture 17], Bass conjectured that ${ \sf d}(G_{ m,n,s}) = m+n-2$ and ${ \sf E}(G_{ m,n,s}) = mn+m+n-2$ provided $ord_n(s) = m$. In this paper, we show that the assumption $ord_n(s) = m$ 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 $r^2 \equiv s \pmod n$.
Categories: math.NT