Center of general linear group is group of scalar matrices over ...

Let GL(n,R) denote the group of n \times n invertible matrices over R . The center of GL(n,R) is the subgroup comprising scalar matrices ... Centerofgenerallineargroupisgroupofscalarmatricesovercenter FromGroupprops Jumpto: navigation, search Template:Sdfcomputation Contents 1Statement 2Proof 2.1Firststep:anyelementofthecentercommuteswithoff-diagonalmatrixunits 2.2Secondstep:Anythingthatcommuteswithoff-diagonalmatrixunitsisadiagonalmatrix 2.3Thirdstep:Anydiagonalmatrixthatcommuteswithallpermutationmatricesisscalar 2.4Combiningthefacts Statement Letbea(notnecessarilycommutative)unitalring,andbeanaturalnumber.Letdenotethegroupofinvertiblematricesover.Thecenterofisthesubgroupcomprisingscalarmatriceswhosescalarentryisacentralinvertibleelementof. Inparticular,forafield,thecentercomprisesscalarmatriceswithanonzeroscalarvalue. Proof Notethattheproofisclearfor,soweconsiderthecasehere. Firststep:anyelementofthecentercommuteswithoff-diagonalmatrixunits Supposearedistinctelementsofand.Definetobethematrixwithintheentryandzeroeselsewhere.istermedthematrixunit. Defineasthesumoftheidentitymatrixand. Notethat: andaretwo-sidedmultiplicativeinversesforany.Thus,. Anymatrixthatcommuteswithmustalsocommutewith,becauseofdistributivityandthefactthatthematrixcommuteswiththeidentity.Thus,anymatrixinthecenterofcommuteswithfor. Secondstep:Anythingthatcommuteswithoff-diagonalmatrixunitsisadiagonalmatrix Supposeisamatrixwithforsome.Considerthematrix.Then,theentryofisnonzero,whiletheentryofiszero.Thus,anymatrixthatcommuteswithalltheoff-diagonalmatrixunitscannothaveanyoff-diagonalentries. Thirdstep:Anydiagonalmatrixthatcommuteswithallpermutationmatricesisscalar Supposeisadiagonalmatrixwith.Thendoesnotcommutewiththepermutationmatrixcorrespondingtothetranspositionofand,becauseconjugationbythatmatrixswitcheswith. Combiningthefacts Combiningthefirsttwostepsyieldsthatanymatrixinthecenterofmustbediagonal,andthethirdstepthenyieldsthatitmustbescalar.Lookingatwhentwoscalarmatricescommute,weseethatthematrixmustinfactbeascalarmatrixwiththescalarvalueitselfacentralandinvertibleelementof. Retrievedfrom"https://groupprops.subwiki.org/w/index.php?title=Center_of_general_linear_group_is_group_of_scalar_matrices_over_center&oldid=50692" Navigationmenu Personaltools Login Namespaces Page Discussion Variants Views Read Viewsource Viewhistory More Search Keylinks GrouppropsmainpageReporterrors/viewlogFAQ1-questionsurveysSubwikirefguideCredits(logo,tech) Lookup Terms/definitionsFacts/theoremsSurveyarticlesSpecificinformation Toptermstoknow SubgroupAbeliangroupTrivialgroupOrderofagroupGeneratingsetHomomorphismIsomorphicgroupsCyclicgroupNormalsubgroupFinitegroupAllbasicdefinitions Populargroups Symmetricgroup:S3(order3!=6)Symmetricgroup:S4(order4!=24)Alternatinggroup:A4(order4!/2=12)Dihedralgroup:D8(order8)Symmetricgroup:S5(order5!=120)Alternatinggroup:A5(order5!/2=60)Quaterniongroup(order8)Allparticulargroups(longlist) Tools WhatlinkshereRelatedchangesSpecialpagesPrintableversionPermanentlinkPageinformation Thispagewaslasteditedon7July2019,at14:43. ContentisavailableunderAttribution-ShareAlike3.0Unportedunlessotherwisenoted. Privacypolicy AboutGroupprops Disclaimers Mobileview