A Characterization of Procyclic Groups via Complete Exterior Degree

Bernardo G. Rodrigues and Francesco G. Russo ()
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Bernardo G. Rodrigues: Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Pretoria 0028, South Africa
Francesco G. Russo: Department of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X1, Rondebosch, Cape Town 7701, South Africa

Mathematics, 2024, vol. 12, issue 7, 1-13

Abstract: We describe the nonabelian exterior square G ∧ ^ 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 ∧ ^ G and new structural results for G ∧ ^ 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.

Keywords: nonabelian exterior square; pro-p-groups; Schur multiplier; free profinite groups (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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