Order of General and Special Linear Group - Cheenta

Here is the post in which you would learn about the Order of General and Special Linear Group with the help of a problem. Try it and learn the solution. OrderofGeneralandSpecialLinearGroupLetpbeaprimenumber.LetGbethegroupofallmatricesoverwithdeterminant1undermatrixmultiplication.ThentheorderofGis(p-1)p(p+1);(AdditionalProblem-thisdidnotcomeinIITJAM2013):Findtheorderofgenerallineargroup.Solution:FirstwefindtheorderofGeneralLinearGroup;itisthegroupofallmatriceswithelementsfromaprimefield(matrixmultiplicationdefinedaccordingly)whichareinvertible.haspelements(0,1,2,...,p-1)hencetobuilda matrixtherearepchoicesforeachofthe4spotscreatingmatricesintotal.Sinceinvertiblematriceshavenonzerodeterminant,wedeductthenumberofmatriceswithzerodeterminantfromtogettheorderof.Supposeamatrixwithzerodeterminantisrepresentedby$latexOtherusefullinks:-https://www.cheenta.com/fermat-point/https://www.youtube.com/watch?v=zEIAqPQnoVk&pbjreload=10RelatedLetpbeaprimenumber.LetGbethegroupofallmatricesoverwithdeterminant1undermatrixmultiplication.ThentheorderofGis(p-1)p(p+1);(AdditionalProblem-thisdidnotcomeinIITJAM2013):Findtheorderofgenerallineargroup.Solution:FirstwefindtheorderofGeneralLinearGroup;itisthegroupofallmatriceswithelementsfromaprimefield(matrixmultiplicationdefinedaccordingly)whichareinvertible.haspelements(0,1,2,...,p-1)hencetobuilda matrixtherearepchoicesforeachofthe4spotscreatingmatricesintotal.Sinceinvertiblematriceshavenonzerodeterminant,wedeductthenumberofmatriceswithzerodeterminantfromtogettheorderof.Supposeamatrixwithzerodeterminantisrepresentedby$latexOtherusefullinks:-https://www.cheenta.com/fermat-point/https://www.youtube.com/watch?v=zEIAqPQnoVk&pbjreload=10RelatedLeaveaReplyCancelreplyLoginwithGoogleThissiteusesAkismettoreducespam.Learnhowyourcommentdataisprocessed.KnowledgePartnerCheentaisaknowledgepartnerofAdityaBirlaEducationAcademyCheentaAcademyAdityaBirlaEducationAcademyCheenta.PassionforMathematicsAdvancedMathematicalScience.Taughtbyolympians,researchersandtruemastersofthesubject.JOINTRIALCOMMUNITYHALLOFFAMESUPERSTARSBOSEOLYMPIADBLOGINFOCUSISI-CMIMODELTESTSSTUDENTOPPORTUNITYGENIUSAPPCHEENTAONDEMANDMORECAREERTEAMCheentaAcademyPrivateLimited|[email protected]