Let p be an odd prime. This dissertation uses the p-exponent lower central series to define the concept of [special characters omitted]-free resolutions of p-groups. A [special characters omitted]-free resolution of a p-group G is a torsion free nilpotent group G¯ together with a epimorphism f : G¯ → G satisfying certain conditions. Then BG¯ is a manifold, therefore the structure of cohomology of G¯ is simpler than that of G. We will show that any p-group has a [special characters omitted]-free resolution. In the case of the unipotent group U( n; [special characters omitted]), an example of [special characters omitted]-free resolution is U(n; [special characters omitted]). Although a [special characters omitted]-free resolution G¯ of a p-group G is not unique, we show that when G has p-nilpotency degree 2, H*(G¯; [special characters omitted]) is independent of choices of G¯. Finally, we calculated cohomology of a family of [special characters omitted]-free resolutions V¯_n for some p-groups V_n and show that the cohomology map H*(V_n; [special characters omitted]) → H*(V¯_n; [special characters omitted]) is onto.