广西自然科学基金资助项目

1、PAGE PAGE 7Appllicaatioon oofThe Funndammenttal Hommomoorphhismm Thheorremof GGrouupLI QQiann-qiian LIIU ZZhi-ganng YANNG LLi-yyingg（Depparttmennt oof MMathhemaaticcs aand Commputter Sciiencce, Guaangxxi TTeaccherrs EEduccatiion Uniiverrsitty,Nannningg Guuanggxi 53300001, P.R.CChinna）Absttracct: The

2、e fuundaamenntall hoomommorpphissm ttheooremm iss veery impporttantt coonseequeencee inn grroupp thheorry, by usiing it we cann reesollve manny pprobblemms. In thiis ppapeer wwe rreseearcchess maainlly aabouut tthe funndammenttal hommomoorphhismm thheorrem appplieed tto ddireect prooduccts of grooup

3、ss annd ggrouup oof iinneer aautoomorrphiismss off a grooup G. Keywwordd: Thhe FFunddameentaal HHomoomorrphiism Theeoreem; Dirrectt Prroduuctss; IInneer AAutoomorrphiismssMR(220033) SSubjjectt Cllasssifiicattionn: 166WChinnesee Liibraary Claassiificcatiion: O1553.33Docuumennt ccodee: A In tthe reaal

4、m of absstraact alggebrra, grooup is onee off thhe bbasiic aand impporttantt coonceept, haave exttenssivee apppliicattionn inn thhe mmathh ittsellf aand manny ssidee off modeern sciiencce ttechhniqque. Foor eexammplee Thheorriess phhysiics, Quuanttum mecchannicss, Quuanttum cheemisstryy, Crrysttalll

5、ogrraphhy aappllicaatioon aare cleear cerrtifficaatioons. Soo thhat, affterr wee sttudyy abbstrractt allgebbra couursee, ggo ddeepp innto a ggrouund of theeoriies of ressearrch to havve tthe neccesssityy veery mucch mmoree. IIn tthe conntennts of grooup, thhe ffunddameentaal hhomoomorrphiism theeore

6、em iis vveryy immporrtannt ttheooremm, wwe ccan usee itt prrovee maany proobleems aboout grooup theeoryy, iin tthiss paaperr too prrovee seeverral conncluusioons as follloww wiith thee fuundaamenntall hoomommorpphissm ttheooremm: TThese coonteentss arre aall staandaard if we nott too thhe sspecciall

7、 prroviisioon aand expplaiinedd.Defiinittionn 1. Leet bbe aa suubgrroupp off a grooup witth ssymbbol , wwe ssay iss thhe nnormmal subbgrooup of iff onne oof tthe folllowwingg coondiitioons holld. To simmpliify mattterrs, we wriite . (1) foor aany ;(2) wheenevver anyy ;(3) foor eeverry andd anny .Def

8、iinittionn 2. Thhe kkernnel of a ggrouup hhomoomorrphiism frrom to a ggrouup witth iidenntitty is thee seet . Thhe kkernnel of iss deenotted by .Defiinittionn 3. Leet bbe aa coolleectiion of grooupss. TThe extternnal dirrectt prroduuct of , 广西自然然科学基基金(0044770388)资助助项目writttenn ass , is thee seet oof

9、 aall m-ttuplees ffor whiich thee itts ccompponeent is an eleemennt oof , annd tthe opeerattionn iss coompoonenntwiise. Inn syymbools =,wherre iis ddefiinedd too bee Notiice thaat iit iis eeasiily to verrifyy thhat thee exxterrnall diirecct pprodductt off grroupps iis iitseelf a ggrouup.4 Defiinitti

10、onn 4. Leet bbe aa grroupp annd bbe aa suubgrroupp off . Forr anny , thhe sset iss caalleed tthe lefft ccoseet oof iin cconttainningg . Anaaloggoussly iss caalleed tthe rigght cosset of H iin conntaiininng .Lemmma 11.11 ( Thhe ffunddameentaal hhomoomorrphiism theeoreem) Leet be a ggrouup hhomoomorrp

11、hiism froom to . TThenn thhe = iss thhe nnormmal subbgrooup of , aand . Too siimpllifyy maatteers, wee caall thee thheorrem as thee FHHT.Lemmma 22.22 Lett bbe aa grroupp hoomommorpphissm ffromm tto . Thhen we havve tthe folllowwingg prropeertiies:（1）IIf iis aa suubgrroupp off , theen iis aa suubgrro

12、upp off ;（2）IIf iis aa noormaal iin, theen iis aa noormaal iin;（3）IIf iis aa suubgrroupp off , theen is a ssubggrouup oof ;（4）IIf iis aa noormaal ssubggrouup oof , thhen iss a norrmall suubgrroupp off Lemmma 33.33 Lett be a hhomoomorrphiism froom a ggrouup tto aa grroupp , aand,. TThenn .Lemmma 44.4

13、4 Lett H be a ssubggrouup oof GG annd llet bellongg too G, thhen:（1） if andd onnly if ;（2） if andd onnly if .By uusinng tthe aboove lemmmass wee caan oobtaain thee foolloowinng mmainnly ressultts.Theooremm 1. Leet GG annd HH bee twwo ggrouups. Suuppoose JG andd KHH, tthenn annd .Prooof. Firrst we wi

14、lll pprovve . FFor anyy aand eveery . WWe hhavee:.Sincce andd , we cann geet , i.e. .Thuss . WWe mmakee usse oof tthe FHTT too prrovee thhat iss issomoorphhic to. Thhereeforre wwe mmustt loook forr a grooup hommomoorphhismm frrom onnto annd ddeteermiine thee keerneel oof iit. In facct oone cann deef

15、inne ccorrrespponddenccedeefinned by . CCleaarlyy, , thheree muust be too saatissfy. Thhus, iss onnto.Becaausee off JGG, wwe hhavee foor, simmilaarlyy, ffor .Whenn , theere aree .For anyy , we havve =.Hencce . Thhereeforre iis ggrouup aa hoomommorpphissm ffromm onntoaand is thee iddenttityy off. For

16、 anyy , tthenn, aaccoordiing to thee prropeertyy off coosett, wwe ccan gett: if andd onnly if annd , i.e. =. Now lett wee loook at ourr prrooff: , iss a grooup hommomoorphhismm frrom ontto andd thhe kkernnel of is . AAccoordiing to thee FHHT, we cann geet .Theooremm 2. LLet iss a grooup hommomoorphh

17、ismm frrom onnto .IIf andd , theen wheere .Prooof: Acccorrdinng tto LLemmma 22.22 (2), wwe kknoww .To eestaabliish , wwe ffirsstlyy neeed to connstrructt a mapppinng andd prrovee iis aa grroupp hoomommorpphissm ffromm oontoo . We ggivee thhe mmapppingg deffineed bby wheere =.For , ssincce is a ssurj

18、jecttionn frrom too , we musst bbe ffounnd ssuchh thhat .TThuss iis oontoo.For arbbitrraryy , Therrefoore iss a grooup hommomoorphhismm.We wwilll noow sshoww , in facct wwe kknoww thhat iss iddenttityy off , acccorddingg too Leemmaa 4, wee caan gget thaat ffor, thhen , ssay , sso tthatt. On thee ott

19、herr haand , , tthatt iss too saay , .MMoreeoveer , beecauuse of , theerefforee . Thaat iis . Acccorrdinng tto tthe FHTT, wwe ccan obttainn .Theeoreem 11 annd TTheooremm 2 appply Exeerciise 1 aand Exeerciise 2.Exerrcisse 11. iss noormaal ssubggrouup oof , iss a norrmall suubgrroupp off .So tthatt fo

20、or aany annd , foor aa fuuncttionn: wee haave iss a grooup isoomorrphiism, soo thhat Assuume andd arre ssetss off alll tthe nonnzerro rreall nuumbeers andd poosittivee reeal nummberrs rresppecttiveely, itt iss reeadiily to verrifyy thhat theey aare inddeedd grroupp wiith orddinaary mulltippliccatiio

21、n.Exerrcisse 22. LLet be genneraal llineear grooup of 22 mmatrricees ooverr unnderr orrdinnaryy maatriix mmulttipllicaatioon . Thhen thee maappiing iss a grooup hommomoorphhismm frrom onnto . TThe grooup off mmatrricees wwithh deeterrminnantt 1 oveer is a nnormmal subbgrooup of . MMoreeoveer .Defiin

22、ittionn 5. Ann auutommorpphissm oof ggrouupiss juust a ggrouup iisommorpphissm ffromm too ittsellf. Thee seet oof aall auttomoorphhismms oof ggrouupiss deenotted by . FFor anyy , is callledd ann innnerr auutommorpphissm oof aand is thee seet oof aall innner auttomoorphhismm off .Theooremm 3: LLetbbe

23、 aa grroupp annd tthe mapppinng deffineed bby . Thhen andd.Prooof. Itt iss cllearrly thaat5.To sshoww , suffficce iit tto pprovve tthatt iis aan aautoomorrphiism of forr anny . 1)(oone-to-onee) FFor anyy , if =, tthenn byy ussingg caanceellaatioon llaw of grooup. Thhus iss onne-tto-oone.2)(oontoo) F

24、For anyy , we takke , thhen, soo thhat iss onnto.3)(OO.P.) FFor anyy , we havve . Thhereeforre is isoomorrphiism froom to .Accoordiing to thee deefinnitiion of auttomoorphhismm. WWe kknoww iis aan aautoomorrphiism of . Notiice thaat ffor anyy , we havve andd .In ffactt foor aany , iit iis ccleaarlyy

25、 . Alsso , Thuss .Sincce , saay , wee haave knoown . WWe ccan obttainn, ii.e. HHencce tthe prooof of is commpleete. It iis eeasyy too seee tthatt foor eeverry, iff annd oonlyy iff whheree iss thhe ccentter of (shhortt foor ).Let bee thhe mmapppingg deefinned by , wwe wwilll prrovee thhat iss a groou

26、p hommomoorphhismm frrom G oontoo I(G) andd thhat C iis iits kerrnell.For eveery , wwe ccan reaadilly ffindd thhat , tthatt iss too saay, is ontto. Forr anny , siincee , so thaat is a ggrouup hhomoomorrphiism froom ontto .Notiice thaat ffor anyy aand eveery , wwe hhavee , i.ee., , tthatt iss . We obttainn, hhencce .