A Characterization of Procyclic Groups via Complete Exterior Degree

We describe the nonabelian exterior square $G \widehat{\wedge} G$ of a pro-$p$-group $G $ (with $p$ arbitrary prime) in terms of quotients of free pro-$p$-groups, providing a new method of construction of $G \widehat{\wedge} G$ and new structural results for $G \widehat{\wedge} G$. Then, we investigate a generalization of the probability that two randomly chosen elements of $G$ commute: this notion is known as the "complete exterior degree" of a pro-$p$-group and we will use it to characterize procyclic groups. Among other things, we present a new formula, which simplifies the numerical aspects which are connected with the evaluation of the complete exterior degree.