Non-existence of universal members in classes of abelian groups

We prove that if μ⁺ < λ = cf(λ) < μ^N0, or if N⁰ < λ < 2^N0, then there is no universal reduced torsion-free abelian group of cardinality λ. We also prove that if (Square original of)_ω < μ⁺ < λ = cf(λ) < μ^N0, then there is no universal reduced separable abelian p-group in λ. We also deal with the class of N₁-free abelian groups. Both results fail if (a) λ = λ^N0 or if (b) λ is a strong limit and cf(μ) = N₀ < μ.