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1、LectureNotesInGroupTheory17maj2007Definition1(Group).Aset{a,b,...}formagroupGif1.Thereexistsancompositionlawa·b=abthatisassociative,i.e.(ab)c=a(bc).2.Thesetisclosedunderthislaw,i.e.ab=c∈G;∀a,b∈G.3.Thereexistsanelemente∈Gwiththepropertyea=a∀a∈Gwhichwecalltheidentity.4.Everyelement
2、a∈Ghasaninverse,denoteda−1∈Gdefinedbyaa−1=e.Remarkthatthecompositionlawdoesnotneedtobecommutative,i.e.ab6=baingeneral.AgroupthatdohaveacommutativecompositionlawarecalledanAbe-liangroup.Thedefinitionforacontinousgroupisthesame,butwewillnotstudycontinousgroupsuntillater.Itshouldals
3、obenotedthatabisauniqueelementinG.Weseethisbylookingatab=ad,ifthiswheretrue,wecanusethecompositionlawandactwiththeinverseofa,a−1fromtheleftonbothsidesandwefindthata−1ab=eb=b=d=ed=a−1ad.Definition2(Subgroup).AsubgroupHofagroupGisasubsetthatitselfformsagroup.Twosubgruopsthatalsoexi
4、stsare{e}andGitself,thesearecalledthetrivialsubgroups.Everyothersubgrupiscalledapropersubgroup.Definition3(Order).TheorderofagroupGisthenumberofelementsinG,denotedg=ord(G).1Groupsofinfiniteordercanbebothinfinetlycountableoruncountable(includingthecontinousgroups).Example1.Groupoft
5、hetriangleTheequilateraltrianglehavethreemirrorsymmetries,mirror-planesthroughthemidd-leofasideanditsoppositevetex,andthreerotationalsymmetries,0,2πand4π.These33symmetriesformthegroupC3v.Ifweconsidertheammoniamolecule(NH3),whichlookslikethistrianglewithhydro-geninitsvertecies,an
6、dnitrogenelevatedinthecenter.Theammoniamoleculeishowevernotsymmetricwithrotationπ(whenwe’flip’themoleculeover)becauseofthenitrogen.SothegroupoftheammoniamoleculeisactuallyD3.Wecanrepresenttherotationalsymmetriesbye(a,b,c)=(a,b,c)(0-rotation)2c3(a,b,c)=(c,a,b)(π-rotation)324c3(a,b
7、,c)=(b,c,a)(π-rotation).3Andthemorrorsymmetriesbyσa(a,b,c)=(a,c,b)σb(a,b,c)=(c,b,a)σc(a,b,c)=(b,a,c).Thisnotationσaissomewhatdevious,itdoesn’tmorroronthevertexabutratheronthefirstelementofthesequence(i,j,k).Itshouldalsobekeptinmindthatthelabe-lingoftheverteciesareanabstractlabeli
8、ng.Ifwedidlabelthethevertecieswewoulddestroythe
