On Bass' conjecture of the small Davenport constant

TL;DR

This paper confirms Bass' conjecture for a class of finite groups and characterizes the structure of extremal product-one free sequences, advancing understanding of the small Davenport constant.

Contribution

It proves Bass' conjecture for groups where the parameter s has order m modulo all prime divisors of n, and characterizes extremal sequences.

Findings

Confirmed Bass' conjecture for specific groups

Characterized structure of extremal product-one free sequences

Generalized previous theorems on the small Davenport constant

Abstract

Let $G$ be a finite group. The small Davenport constant $\mathsf d(G)$ of $G$ is the maximal integer $\ell$ such that there is a sequence of length $\ell$ over $G$ which has no nonempty product-one subsequence. In 2007, Bass conjectured that $\mathsf d(G_{m,n})=m+n-2$, where $G_{m,n}=\langle x, y| x^m=y^n=1, x^{-1}yx=y^s\rangle$, and $s$ has order $m$ modulo $n$. In this paper, we confirm the conjecture for any group $G_{m,n}$ with additional conditions that $s$ has order $m$ modulo $q$, for every prime divisor $q$ of $n$. Moreover, we solve the associated inverse problem characterizing the structure of any product-one free sequence with extremal length $\mathsf d(G_{m,n})$. Our results generalize some obtained theorems on this problem.