For an infinite, finitely generated group Γ, we study the first cohomology group H1(Γ,λΓ) with coefficients in the left regular representation λΓ of Γ on l2(Γ). We first prove thatH Γ(Γ, C Γ) embeds into HΓ(Γ,λΓ); as a consequence, ifH Γ(Γ,λΓ)=0, then Γ is not amenable with one end. For a Cayley graph X of Γ, denote by HD(X) the space of harmonic functions on X with finite Dirichlet sum. We show that, if Γ is not amenable, then there is a natural isomorphism betweenH Γ(Γ,λΓ) and \(HD(X)/\mathbb{C} \) (the latter space being isomorphic to the first Ll-cohomology space of Γ). We draw the following consequences: