Formation process of ferromagnetic nanodot arrays on a periodically nanopatterned Cu(001) surface with nitrogen atoms are overviewed, and their magnetic properties studied by surface magneto-optical Kerr effect (SMOKE) in ultra high vacuum are described. The square lattice nanopattern induced by the strain due to nitrogen adsorption is formed by the strain-relief mechanism of the Cu lattice near the surface. On top of that, the diffusion and the adsorption of the deposited atoms depend on the substrate lattice strain, and the ferromagnetic metal island preferentially grow at the less strained clean Cu surface. As a result, a square nanodot array of 7 nm × 7 nm is formed on the surface. The hysteresis loops of the arrays were in-situ measured by SMOKE between 400 K and 95 K. The results indicate long range ferromagnetic order is established among the superparamagnetic Co dots of two-monoatomic-layers high by the magnetic interaction through the monolayer Co strips.