A note on Bass? conjecture

For a finite group G, we denote by d(G) and by 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 Ns Cm be a metacyclic group. In [2, Conjecture 17], Bass conjectured that d(Gm,n,s) = m + n - 2 and 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-pro duct-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G2m,2n,r, where r2 equivalent to s (mod n).(c) 2023 Elsevier Inc. All rights reserved.