Let λ be an infinite cardinal. Set λ0 = λ, define λi+1 = 2λi for every i = 0, 1,..., take µ as the first cardinal with λi < µ, i = 0, 1,... and put κ = (µℵ0 ) +. If F is a torsion-free group of cardinality at least κ and K is its subgroup such that F/K is torsion and |F/K| ≤ λ, then K contains a non-zero subgroup pure in F. This generalizes the result from a previous paper dealing with F/K p-primary.