# A note on Bass' conjecture

## TL;DR

This paper examines Bass' conjecture on the Davenport and Gao constants for certain non-abelian groups, demonstrating the necessity of specific order conditions and exploring implications for related groups.

## Contribution

It shows that the order condition in Bass' conjecture is essential and extends the conjecture's implications to related metacyclic groups under certain assumptions.

## Key findings

- The order condition $ord_n(s) = m$ is necessary for Bass' conjecture.
- Bass' conjecture holds under the assumption of well-behaved maximal length product-one free sequences.
- Implications for groups $G_{2m,2n,r}$ are established if the conjecture holds for $G_{m,n,s}$.

## Abstract

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 $r^2 \equiv s \pmod n$.