On graphs related to conjugacy classes of groups
Identifieur interne : 000D58 ( Istex/Corpus ); précédent : 000D57; suivant : 000D59On graphs related to conjugacy classes of groups
Auteurs : Guy AlfandarySource :
- Israel Journal of Mathematics [ 0021-2172 ] ; 1994-10-01.
English descriptors
- Teeft :
Abstract
Abstract: LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible.
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DOI: 10.1007/BF02773678
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Math.</title><idno type="ISSN">0021-2172</idno><idno type="eISSN">1565-8511</idno><imprint><publisher>Springer-Verlag</publisher><pubPlace>New York</pubPlace><date type="published" when="1994-10-01">1994-10-01</date><biblScope unit="volume">86</biblScope><biblScope unit="issue">1-3</biblScope><biblScope unit="page" from="211">211</biblScope><biblScope unit="page" to="220">220</biblScope></imprint><idno type="ISSN">0021-2172</idno></series></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">0021-2172</idno></seriesStmt></fileDesc><profileDesc><textClass><keywords scheme="Teeft" xml:lang="en"><term>Abelian</term><term>Abelian hall</term><term>Abelian kernel</term><term>Complete graph</term><term>Conjugacy</term><term>Conjugacy classes</term><term>Finite group</term><term>Marcel herzog</term><term>Normal hall</term><term>Semidirect decomposition</term><term>Subgroup</term><term>Sylow</term><term>Sylow subgroup</term></keywords></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract" xml:lang="en">Abstract: LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. 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Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible.</abstract><qualityIndicators><refBibsNative>false</refBibsNative><abstractWordCount>177</abstractWordCount><abstractCharCount>1009</abstractCharCount><keywordCount>0</keywordCount><score>7.543</score><pdfWordCount>3419</pdfWordCount><pdfCharCount>13135</pdfCharCount><pdfVersion>1.3</pdfVersion><pdfPageCount>10</pdfPageCount><pdfPageSize>461 x 666 pts</pdfPageSize></qualityIndicators><title>On graphs related to conjugacy classes of groups</title><genre><json:string>research-article</json:string></genre><host><title>Israel Journal of Mathematics</title><language><json:string>unknown</json:string></language><publicationDate>1994</publicationDate><copyrightDate>1994</copyrightDate><issn><json:string>0021-2172</json:string></issn><eissn><json:string>1565-8511</json:string></eissn><journalId><json:string>11856</json:string></journalId><volume>86</volume><issue>1-3</issue><pages><first>211</first><last>220</last></pages><genre><json:string>journal</json:string></genre><subject><json:item><value>Mathematics</value></json:item><json:item><value>Mathematics, general</value></json:item><json:item><value>Algebra</value></json:item><json:item><value>Group Theory and Generalizations</value></json:item><json:item><value>Analysis</value></json:item><json:item><value>Applications of Mathematics</value></json:item><json:item><value>Mathematical and Computational Physics</value></json:item></subject></host><namedEntities><unitex><date></date><geogName></geogName><orgName></orgName><orgName_funder></orgName_funder><orgName_provider></orgName_provider><persName></persName><placeName></placeName><ref_url></ref_url><ref_bibl></ref_bibl><bibl></bibl></unitex></namedEntities><ark><json:string>ark:/67375/1BB-JJZ6M2C1-9</json:string></ark><categories><wos><json:string>1 - science</json:string><json:string>2 - mathematics</json:string></wos><scienceMetrix><json:string>1 - natural sciences</json:string><json:string>2 - mathematics & statistics</json:string><json:string>3 - general mathematics</json:string></scienceMetrix><scopus><json:string>1 - Physical Sciences</json:string><json:string>2 - Mathematics</json:string><json:string>3 - General Mathematics</json:string></scopus><inist><json:string>1 - sciences appliquees, technologies et medecines</json:string><json:string>2 - sciences exactes et technologie</json:string><json:string>3 - sciences et techniques communes</json:string><json:string>4 - mathematiques</json:string></inist></categories><publicationDate>1994</publicationDate><copyrightDate>1994</copyrightDate><doi><json:string>10.1007/BF02773678</json:string></doi><id>A1FAA67F41C80130B50E1266E233CC566AB1D4B6</id><score>1</score><fulltext><json:item><extension>pdf</extension><original>true</original><mimetype>application/pdf</mimetype><uri>https://api.istex.fr/ark:/67375/1BB-JJZ6M2C1-9/fulltext.pdf</uri></json:item><json:item><extension>zip</extension><original>false</original><mimetype>application/zip</mimetype><uri>https://api.istex.fr/ark:/67375/1BB-JJZ6M2C1-9/bundle.zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/ark:/67375/1BB-JJZ6M2C1-9/fulltext.tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">On graphs related to conjugacy classes of groups</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher scheme="https://scientific-publisher.data.istex.fr">Springer-Verlag</publisher><pubPlace>New York</pubPlace><availability><licence><p>The Magnes Press, 1994</p></licence><p scheme="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-3XSW68JL-F">springer</p></availability><date>1993-05-03</date></publicationStmt><notesStmt><note type="research-article" scheme="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</note><note type="journal" scheme="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</note></notesStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">On graphs related to conjugacy classes of groups</title><author xml:id="author-0000" corresp="yes"><persName><forename type="first">Guy</forename><surname>Alfandary</surname></persName><affiliation>School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel</affiliation></author><idno type="istex">A1FAA67F41C80130B50E1266E233CC566AB1D4B6</idno><idno type="ark">ark:/67375/1BB-JJZ6M2C1-9</idno><idno type="DOI">10.1007/BF02773678</idno><idno type="article-id">BF02773678</idno><idno type="article-id">Art7</idno></analytic><monogr><title level="j">Israel Journal of Mathematics</title><title level="j" type="abbrev">Israel J. 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Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible.</p></abstract><textClass><keywords scheme="Journal-Subject-Group"><list><label>M</label><item><term>Mathematics</term></item><label>M00009</label><item><term>Mathematics, general</term></item><label>M11000</label><item><term>Algebra</term></item><label>M11078</label><item><term>Group Theory and Generalizations</term></item><label>M12007</label><item><term>Analysis</term></item><label>M13003</label><item><term>Applications of Mathematics</term></item><label>P19005</label><item><term>Mathematical and Computational Physics</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1993-05-03">Created</change><change when="1994-10-01">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/ark:/67375/1BB-JJZ6M2C1-9/fulltext.txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus springer-journals not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="UTF-8"</istex:xmlDeclaration><istex:docType PUBLIC="-//Springer-Verlag//DTD A++ V2.4//EN" URI="http://devel.springer.de/A++/V2.4/DTD/A++V2.4.dtd" name="istex:docType"></istex:docType><istex:document><Publisher><PublisherInfo><PublisherName>Springer-Verlag</PublisherName><PublisherLocation>New York</PublisherLocation></PublisherInfo><Journal><JournalInfo JournalProductType="ArchiveJournal" NumberingStyle="Unnumbered"><JournalID>11856</JournalID><JournalPrintISSN>0021-2172</JournalPrintISSN><JournalElectronicISSN>1565-8511</JournalElectronicISSN><JournalTitle>Israel Journal of Mathematics</JournalTitle><JournalAbbreviatedTitle>Israel J. 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Attach to<Emphasis Type="Italic">G</Emphasis> the following two graphs: Γ — its vertices are the non-central conjugacy classes of<Emphasis Type="Italic">G</Emphasis>, and two vertices are connected if their sizes are<Emphasis Type="Italic">not</Emphasis> coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes of<Emphasis Type="Italic">G</Emphasis>, and two vertices are connected if they both divide the size of some conjugacy class of<Emphasis Type="Italic">G</Emphasis>. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only if<Emphasis Type="Italic">G</Emphasis> is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only if<Emphasis Type="Italic">G</Emphasis> is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. In [2],[3] it is shown that the above bounds are best possible.</Para></Abstract><ArticleNote Type="Misc"><SimplePara>The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at the Tel Aviv University under the supervision of Prof. Marcel Herzog.</SimplePara></ArticleNote></ArticleHeader><NoBody></NoBody></Article></Issue></Volume></Journal></Publisher></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>On graphs related to conjugacy classes of groups</title></titleInfo><titleInfo type="alternative" contentType="CDATA"><title>On graphs related to conjugacy classes of groups</title></titleInfo><name type="personal" displayLabel="corresp"><namePart type="given">Guy</namePart><namePart type="family">Alfandary</namePart><affiliation>School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="OriginalPaper" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Springer-Verlag</publisher><place><placeTerm type="text">New York</placeTerm></place><dateCreated encoding="w3cdtf">1993-05-03</dateCreated><dateIssued encoding="w3cdtf">1994-10-01</dateIssued><copyrightDate encoding="w3cdtf">1994</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><abstract lang="en">Abstract: LetG be a finite group. Attach toG the following two graphs: Γ — its vertices are the non-central conjugacy classes ofG, and two vertices are connected if their sizes arenot coprime, and Γ* — its vertices are the prime divisors of sizes of conjugacy classes ofG, and two vertices are connected if they both divide the size of some conjugacy class ofG. We prove that whenever Γ* is connected then its diameter is at most 3, (this result was independently proved in [3], for solvable groups) and Γ* is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. Using the method of that proof we give an alternative proof to Theorems in [1],[2],[6], namely that the diameter of Γ is also at most 3, whenever the graph is connected, and that Γ is disconnected if and only ifG is quasi-Frobenius with abelian kernel and complements. As a result we conclude that both Γ and Γ* have at most two connected components. 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