广西自然科学基金资助项目

1、PAGE PAGE 10Application ofThe Fundamental Homomorphism Theoremof GroupLI Qiian-qiaan LIUU Zhhi-ggangg YYANGG Lii-yiing（Depaartmmentt off Maatheematticss annd CCompputeer SScieencee, GGuanngxii Teeachherss Edducaatioon UUnivverssityy,Nanniing Guaangxxi 53000011, P.RR.Chhinaa）Abstrractt: TThe funndamm

2、enttal hommomoorphhismm thheorrem is verry iimpoortaant connseqquennce in grooup theeoryy, bby uusinng iit wwe ccan ressolvve mmanyy prrobllemss. IIn tthiss paaperr wee reeseaarchhes maiinlyy abboutt thhe ffunddameentaal hhomoomorrphiism theeoreem aapplliedd too diirecct pprodductts oof ggrouups and

3、d grroupp off innnerr auutommorpphissms of a ggrouup GG. Keywoord: Thhe FFunddameentaal HHomoomorrphiism Theeoreem; Dirrectt Prroduuctss; IInneer AAutoomorrphiismssMR(20003) Suubjeect Claassiificcatiion: 166WChineese Libbrarry CClasssifficaatioon: O1553.33Docummentt coode: AA In thhe rreallm oof aab

4、sttracct aalgeebraa, ggrouup iis oone of thee baasicc annd iimpoortaant connceppt, havve eexteensiive apppliccatiion in thee maath itsselff annd mmanyy siide of modeern sciiencce ttechhniqque. Foor eexammplee Thheorriess phhysiics, Quuanttum mecchannicss, Quuanttum cheemisstryy, Crrysttalllogrraphhy

5、 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 theeoreem iis vv

6、eryy 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 prroviis

7、ioon aand expplaiinedd.Definnitiion 1. Lett bee a subbgrooup of a ggrouup wwithh syymbool , wwe ssay iss thhe nnormmal subbgrooup of iff onne oof tthe folllowwingg coondiitioons holld. To simmpliify mattterrs, we wriite . (1) forr anny ;(2) wwhenneveer aany ;(3) forr evveryy aand anyy .Definnitiion

8、2. Thee keerneel oof aa grroupp hoomommorpphissm froom tto aa grroupp wwithh iddenttityy iis tthe sett . Thee keerneel oof is dennoteed bby .Definnitiion 3. Lett bee a colllecctioon oof ggrouups. Thhe eexteernaal ddireect prooducct oof , 广西自然科科学基金金(044470038)资助项项目writtten as , iis tthe sett off alll

9、 mm-tuplees ffor whiich thee itts ccompponeent is an eleemennt oof , annd tthe opeerattionn iss coompoonenntwiise. Inn syymbools =,wheree iss deefinned to be Noticce tthatt itt iss eaasilly tto vveriify thaat tthe extternnal dirrectt prroduuct of grooupss iss ittsellf aa grroupp.44 Definnitiion 4. L

10、ett bee a grooup andd bee a subbgrooup of . Forr anny , thhe sset iss caalleed tthe lefft ccoseet oof iin cconttainningg . Anaaloggoussly iss caalleed tthe rigght cosset of H iin conntaiininng .Lemmaa 1.1 ( Thhe ffunddameentaal hhomoomorrphiism theeoreem) Leet be a ggrouup hhomoomorrphiism froom to

11、. TThenn thhe = iss thhe nnormmal subbgrooup of , aand . To simmpliify mattterrs, we calll tthe theeoreem aas tthe FHTT.Lemmaa 2.2 LLet bee a grooup hommomoorphhismm frrom too . Theen wwe hhavee thhe ffolllowiing prooperrtiees:（1）Iff iss a subbgrooup of , tthenn iss a subbgrooup of ;（2）Iff iss a nor

12、rmall inn, tthenn iss a norrmall inn;（3）Iff iss a subbgrooup of , tthenn iis aa suubgrroupp off ;（4）Iff iss a norrmall suubgrroupp off , theen is a nnormmal subbgrooup of Lemmaa 3.3 LLet be a hhomoomorrphiism froom a ggrouup tto aa grroupp , aand,. TThenn .Lemmaa 4.4 Leet HH bee a subbgrooup of G aa

13、nd lett beelonng tto GG, tthenn:（1） iif aand onlly iif ;（2） iif aand onlly iif .By ussingg thhe aabovve llemmmas we cann obbtaiin tthe folllowwingg maainlly rresuultss.Theorrem 1. Lett G andd H be twoo grroupps. Suppposse JJG aand KH, thhen andd .Prooff. FFirsst wwe wwilll prrovee . Foor aany annd e

14、everry . Wee haave:.Sincee aand , wwe ccan gett , i.ee. .Thus . Wee maake usee off thhe FFHT to proove thaat is isoomorrphiic tto. Theerefforee wee muust loook ffor a ggrouup hhomoomorrphiism froom ontto andd deeterrminne tthe kerrnell off itt. IIn ffactt onne ccan deffinee coorreespoondeenceedeffin

15、eed bby . Cllearrly, , theere musst bbe to sattisffy. Thuus, is ontto.Becauuse of JG, wee haave forr, ssimiilarrly, foor .When , ttherre aare .For aany , wwe hhavee =.Hencee . Theerefforee iss grroupp a hommomoorphhismm frrom onttoannd iis tthe ideentiity of. For aany , tthenn, aaccoordiing to thee

16、prropeertyy off coosett, wwe ccan gett: if andd onnly if annd , i.e. =. Now llet we loook aat oour prooof: , is a ggrouup hhomoomorrphiism froom oontoo aand thee keerneel oof iis . Acccorrdinng tto tthe FHTT, wwe ccan gett .Theorrem 2. Leet is a ggrouup hhomoomorrphiism froom ontto .Iff aand , tthen

17、n wwherre .Prooff: Acccorddingg too Leemmaa 2.2 (2), wee knnow .To esstabblissh , wee fiirsttly neeed tto cconsstruuct a mmapppingg aand proove iss a grooup hommomoorphhismm frrom onnto . We giive thee maappiing deffineed bby wheere =.For , siincee iis aa suurjeectiion froom to , wwe mmustt bee foou

18、ndd suuch thaat .Thhus iss onnto.For aarbiitraary , Thereeforre is a ggrouup hhomoomorrphiism.We wiill noww shhow , iin ffactt wee knnow thaat is ideentiity of , aaccoordiing to Lemmma 4, we cann geet tthatt foor, theen , saay , soo thhat. On tthe othher hannd , , tthatt iss too saay , .MMoreeoveer

19、, beccausse oof , ttherrefoore . TThatt iss . Acccorddingg too thhe FFHT, wee caan oobtaain .Theooremm 1 andd Thheorrem 2 aapplly EExerrcisse 11 annd EExerrcisse 22.Exerccisee 1. iss noormaal ssubggrouup oof , iss a norrmall suubgrroupp off .So thhat forr anny andd , forr a funnctiion: wee haave iss

20、 a grooup isoomorrphiism, soo thhat Assumme aand aree seets of alll thhe nnonzzeroo reeal nummberrs aand possitiive reaal nnumbberss reespeectiivelly, it is reaadilly tto vveriify thaat ttheyy arre iindeeed grooup witth oordiinarry mmulttipllicaatioon.Exerccisee 2. Leet bbe ggeneerall liineaar ggrou

21、up oof 222 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 .Definnitiion 5. An auttomoorphhismm off grrouppis ju

22、sst aa grroupp issomoorphhismm frrom to itsselff. TThe sett off alll aautoomorrphiismss off grrouppis dennoteed bby . Foor aany , iss caalleed aan iinneer aautoomorrphiism of andd iss thhe sset of alll innnerr auutommorpphissm oof .Theorrem 3: Leetbee a grooup andd thhe mmapppingg deffineed bby . Th

23、hen andd.Prooff. It is cleearlly tthatt5.To shhow , ssuffficee itt too prrovee thhat iss ann auutommorpphissm oof ffor anyy . 1)(onne-tto-oone) Foor aany , iif =, tthenn byy ussingg caanceellaatioon llaw of grooup. Thhus iss onne-tto-oone.2)(onnto) Foor aany , wwe ttakee , theen, so thaat is ontto.3

24、)(O.P.) Foor aany , wwe hhavee . Theerefforee iis iisommorpphissm ffromm tto .Accorrdinng tto tthe deffiniitioon oof aautoomorrphiism. Wee knnow iss ann auutommorpphissm oof . Noticce tthatt foor aany , wwe hhavee aand .In faact forr anny , itt iss cllearrly . AAlsoo , Thus .Sincee , sayy , we havve

25、 kknowwn . Wee caan oobtaain, i.e. Heencee thhe pprooof oof is commpleete. It iss eaasy to seee thhat forr evveryy, iif aand onlly iif wwherre iis tthe cennterr off (sshorrt ffor ).Let be thee maappiing deffineed bby , wee wiill proove thaat is a ggrouup hhomoomorrphiism froom GG onnto I(GG) aand thaat CC iss itts kkernnel.For eeverry , wee caan rreaddilyy fiind thaat , thhat is to sayy, iis oontoo. FFor anyy , sinnce , sso tthatt iis aa grroupp hoomommorpphissm ffromm oontoo .Noticce tthatt foor aany annd eeverry , wee haave , ii.e., , thhat is . WWe oobtaain, heencee . Ne