A note on Bass' conjecture

Abstract

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⋊sCm 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-product-one free sequences of maximal length are well behaved, then Bass conjecture also holds for G2m,2n,r, where r2≡s(modn).