On Bass' conjecture of the small Davenport constant

2025/02/19 by Wang, Guoqing, Zhao, Yang
#11B75 #11P70 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2502.13409

Abstract

Let G be a finite group. The small Davenport constant \mathsf d(G) of G is the maximal integer ℓ such that there is a sequence of length ℓ 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=⟨ x, y| x^m=yⁿ=1, x^-1yx=y^s⟩, 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.

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