There have been a lot of research done on the relationship between locally compact groups and algebras associated with them. For example, Johnson proved that a locally compact group G is amenable if and only if the convolution algebra L₁(G) is amenable as a Banach algebra, and Ruan showed that G is amenable if and only if the Fourier algebra A(G) of G is operator amenable. Motivated by Ruan's work, we want to study G through tools from p-operator spaces. We first introduce the p-operator space and various p-operator space tensor products. We then study p-operator space approximation property and p-operator space completely bounded approximation property which are related to p-operator space injective tensor product. We then apply these properties to the study of the pseudofunction algebra PF_p(G), the pseudomeasure algebra PM_p(G), and the Figà-Talamanca-Herz Algebra A_p( G). Especially we show that if G is discrete, the most of approximation properties for the reduced group C*-algebra [special characters omitted](G), the group von Neumann algebra VN( G), and the Fourier algebra A(G) (related to amenability, weak amenability, and approximation property of G) have natural p-analogues for PF _p(G), PM_p( G), and A_p(G). With help of Herz's work, we also study the stability of these properties. Finally we discuss the properties C_p, [special characters omitted], and [special characters omitted] which are natural p-analogues of properties C, C', and C".