2023/02/23 by Avelar, Danilo Vilela, Martínez, Fabio Enrique Brochero, Ribas, Sávio · 1 citation
#FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2302.11754
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 Cn be the cyclic group of order n and let Gm,n,s = Cn \rtimess Cm 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 \sf d(Gm,n,s) = m+n-2 and \sf E(Gm,n,s) = mn+m+n-2 provided ordn(s) = m. In this paper, we show that the assumption ordn(s) = m is essential and cannot be removed. Moreover, if we suppose that Bass' conjecture holds for Gm,n,s and the mn-product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G2m,2n,r, where r2 ≡ s \pmod n.