A Characterization of Procyclic Groups via Complete Exterior Degree
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&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/7/1018 |
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| author | Bernardo G. Rodrigues Francesco G. Russo |
| author_facet | Bernardo G. Rodrigues Francesco G. Russo |
| author_sort | Bernardo G. Rodrigues |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-24T10:40:38Z |
| format | Article |
| id | doaj.art-e830982e20e3436c90e3c4141722b81a |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-04-24T10:40:38Z |
| publishDate | 2024-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-e830982e20e3436c90e3c4141722b81a2024-04-12T13:22:38ZengMDPI AGMathematics2227-73902024-03-01127101810.3390/math12071018A Characterization of Procyclic Groups via Complete Exterior DegreeBernardo G. Rodrigues0Francesco G. Russo1Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Pretoria 0028, South AfricaDepartment of Mathematics and Applied Mathematics, University of Cape Town, Private Bag X1, Rondebosch, Cape Town 7701, South AfricaWe 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.https://www.mdpi.com/2227-7390/12/7/1018nonabelian exterior squarepro-p-groupsSchur multiplierfree profinite groups |
| spellingShingle | Bernardo G. Rodrigues Francesco G. Russo A Characterization of Procyclic Groups via Complete Exterior Degree Mathematics nonabelian exterior square pro-p-groups Schur multiplier free profinite groups |
| title | A Characterization of Procyclic Groups via Complete Exterior Degree |
| title_full | A Characterization of Procyclic Groups via Complete Exterior Degree |
| title_fullStr | A Characterization of Procyclic Groups via Complete Exterior Degree |
| title_full_unstemmed | A Characterization of Procyclic Groups via Complete Exterior Degree |
| title_short | A Characterization of Procyclic Groups via Complete Exterior Degree |
| title_sort | characterization of procyclic groups via complete exterior degree |
| topic | nonabelian exterior square pro-p-groups Schur multiplier free profinite groups |
| url | https://www.mdpi.com/2227-7390/12/7/1018 |
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