Using the notion of complete nonabelian exterior square G∧ˆG of a pro-p-group G (p prime), we develop the theory of the exterior degree dˆ(G) in the infinite case, focusing on its relations with the probability of commuting pairs d(G). Among the main results of this paper, we describe upper and lower bounds for dˆ(G) with respect to d(G). Here the size of the second homology group H_2(G,Z_p) (over the p-adic integers Z_p) plays a fundamental role. A further result of homological nature is placed at the end, in order to emphasize the influence of H_2(G,Z_p) both on G and dˆ(G).
