We study torsion phenomena in the integral cohomology of finite if-spaces X through the Eilenberg-Moore spectral sequence converging to H＊($Omega$X; Z$_{p}$). We also investigate how the difference between the Z$_{p}$-filtration length f$_{p}$(X) and the Z$_{p}$-cup length c$_{p}$(X) on a simply connected finite H-space X is reflected in the Eilenberg-Moore spectral sequence converging to H＊($Omega$X;Z$_{p}$). Finally we get the following result: Let p be an odd prime and X an n-connected finite H-space with dim QH＊ (X;Z$_{p}$) $leq$ m. Then H＊(X;Z) is p-torsion free if (equation omitted).tion omitted).