| Summary: | We describe the nonabelian exterior square <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mover accent="true"><mo>∧</mo><mo>^</mo></mover><mi>G</mi></mrow></semantics></math></inline-formula> of a pro-<i>p</i>-group <i>G</i> (with <i>p</i> arbitrary prime) in terms of quotients of free pro-<i>p</i>-groups, providing a new method of construction of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mover accent="true"><mo>∧</mo><mo>^</mo></mover><mi>G</mi></mrow></semantics></math></inline-formula> and new structural results for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mover accent="true"><mo>∧</mo><mo>^</mo></mover><mi>G</mi></mrow></semantics></math></inline-formula>. Then, we investigate a generalization of the probability that two randomly chosen elements of <i>G</i> commute: this notion is known as the “complete exterior degree” of a pro-<i>p</i>-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.
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