Discretely normed Abelian groups

A discrete norm on an Abelian groupA is a non-negative function ‖ · ‖A → ℝ which satisfies the triangle inequality, is homogenous with respect to scaling ofA by ℤ and is bounded away from 0 onA/{0}.A countable Abelian group is discretely normed if and only if the group is free.