Luận văn Về nhóm con đa chuẩn tắc của nhóm S5

18 ~ ~ ? ~ CmJdNG2 CACNHOMCONCUANHOMSs 2.1.Mot8619thuye'tc6lienquan: [11,tr26-84] 2.1.1.Djnh1:9(Lagrange)NellGIII nh6mhfluh/jlnvaS:::;G chiIs 11a u'dccua IG 1 va[G:S] ='I~i 2.1.2.M~nhd~. Nellf 1am9tsO"nguyento"va I G I ~p thi G III m9tnh6m cyc]jc. 2.1.3.M~nhd~.NellH:::;G c6chiso"2thia2EH, t1aE G 2.1.4.Bfl d~.Nell G 1anh6mcancycliccapn thit6nt/jlidaynhf{tm9t nh6mcancapd, vdimollfdc d cuan . 2.1.5.Bfld~.Nell G1anR6mthiquailh~ 1/ Y lienhr;fpvdix trongG 1/ dlfr;fc xacd.inhbiJiy =gxg-I, 3 g E G, Ja quailh~tlfdngdlfdng. 2.1.6.Djnhngma.G la nh6m . a EG , aG={xax-llXEG }duQcgQila 1dplienhc;fpcuaph'antli'a . H:::;G , gEG , Hg=gHg-1= {gHg-11h EH }duQcgQila nh6mcan cuaG ]jellhc;fpvaiH nhoph'antifg E G vaHg :::;G . NG(H)={xEG I XHX-I=H}duQcgQilachafinh6atltcuaHtrong G vaNG(H)la nh6mcan1annhatcuaG nh~nH lamnh6mcon chu§nt~c 2.1.7.Djnh1:9.Neu-H:::;G thiso"llfc;fngc cacnh6mconlien hr;fpcuaH trongG blingchiso"cuachafinh6atifcuaH trongG, tdc1a c=[G: NG(H)] vac1alfaccua I G I nellGhuuhfjln. Hdn nua, aHa-I =bHHIkhi vachikhi HIa E NG(H) I L 2.1.8.M~nhd~.Nellnh6mGhiluh?nc6capchiahetchom(JtsO"nguyen t6'pthiG chlia1ph'antiJ'capp. 19 2.1.9.Binh 11(Sylow,1872).G la nh6mhiluh?n, I G I =pmrvajplas6" nguyent6"vakh6ngla lfaccuar . Khj d6. j) ji) jij) - iv) 2.1.10. i) ii) iii) G c6nh6mcancappm.M(Jt nh6mcannhlfV?ydU'r;fCgQila nh6mcanSylowcuaG. MQip-nh6mcancuaG dellnlimtrongm(Jtp-nh6mcanSylow nao d6cuaG. M9ip-nh6mcanSylowcuaGdeujjenhf/pvcJjnhautrangG. GQj n 1i1s6/p-nh6mcanSylowcuaG thin la lfaccua I G I va n ==1 (modp). o Ja nh6m,H::; OJ a E 0, 5 C OJkhi d6: xCG(a)x.I=CG(xax.I) xNG(H)x.I=NG(xHx.I) X X.I = Chungminh. i) Ta coY E xCG(a)x.l<=?X.lyxE CG(a) <=?(x.lyx)a=a (X.lyX) <=?x(x.lyx)ax.l =xa (X.IYX)X.l <=?y(xax.l)=(xax.l)y ii) <=?Y E CG(xax.l) VfJy : xCG(a)x.l=CG(xax.l) Ta coy E xl"{Q(H)x.l<=?X.lyx E NG(H) <=?(x.lyx)H =H (X.lyX) <=?y(x.IHx) =(XHX.I)y <=?Y E NQ(xHx.l) 20 iii) GiiisaS={Sb Sz,...,Sk} ri E ll, k E IN Ta c6y E Xx-Iy =XS~IS;2...S~kX-1 y=(XS;IX-I)(XS;2X-I)...(XS~X-l) y = (XSIX-I)'1(XS2X-l)'2",(XSkX-l)'k Y E Do d6: x X-I= 2.1.11..M~nhd'e.G Iii nh6mhfluh~n,H lap-nh6mcanSylowcilaG.Khj d6,H <1G khj vachikhj G c6duynhatmQtp-nh6mcan Sylow-laH Chungminh: (=»Giii saP lap-nh6mcanSylowcuaG thltheodinh11Sylow3xE G saochoP=xHx-InhungH <1G nenXHX-I=H.SuyfaP=H. «=)H la p-nh6mconSylowcuaGnen'\IxE Gtac6xHx-Iclinglap- nh6mcanSyloweuaG.Dotintduynha'tcuaH nenxHx-I=H. Suy fa H <]G. 2.1.12.M~nhd'e.Kf hj~uSnla nh6mdolxungb$cn , 5 E Sn , 8= 5]52...5k vdj 5], 52,...,5k la cac yangxich d(Jcltip [an lU<;ftco cap la n],n2,"',nkthi 181 =BCNN(n], n2,"',nIJ 2.1.13.Caedinh11diiduQcchungminhtrong[5] . M9j nhomcanAbelcap6d8ucyclk. . M9j nhomcap4ho?cding cauV(JjZ4ho?cding caavdjZ2xZ2. . M9j nhomcap6ho?cding cauV(JjZ6ho?cding cauvdjSJ. . Nh6mthayphjenAn(vdjn~4)du<;fCsjnhra biJj cac3 chutrinh. . Vdjn~3,SncoduynhatmQtnhomcanchiso/2laAn- 21 2.1.14.M~nhd'e.[6,tr34]A4cocap12nhlfngkhongchuanhomcancap6. 2.1.15.M~nhd'e.Nell G cocap12vaGkhangding CallvdiA4 thiG chria ph'antztclf; 6 Hon lilla, G e6 illQt3-nh6illcan Sylow. ehu~nt~e. 2.1.16.B6 d'e.Nell a,j3E Snthi aj3a-i1.1m9tphep the"co clingCalltruc I vongxfchnhlt13,no thudltrjcnhC!tacod9ngcuaa 1entung ki hi~uco trong13. Tric1.1:13=(aia2...aiJE Sn, vdimtiia E Sn, taco: r=aj3a-l=(a(ai)a(a2)."a(aiJ) 2.1.17.DinhIi. a,j3E Snlienhrjpnhaukhi vachjkhi chungc6clingCall trucvongxich.Tli dinh11naytatha'yell2 phepthe'cocling ea'utruevangxichtaluantImduQeYESnd~Y~i1=a. 2.1.18.DinhIi. An1.1nh6madD,'rIn~5 2.1.19.Dinhngma.Nhomdjhedra1D2nvdi n~2laillQtnhomcanea'p2n d . h b?' 2 h'" ? , h? n? 1 ' -1 -1u<;1eSIn ra 01 p an tu Sva t t oa: S =c= va tst =s 2.1.20.DinhIi. 2-nhomcanSylowcuaS5dingCallvdiDs 2.1.21.DinhIi. Nellpnguyenta"thimtiinhomGcap2pho;fic1.1cyclic ho;fic1adihedral. 2.1.22.DinhIi. Giasri I G I =pq,trongaop,q 1acacsa"nguyenta"va p>q.Khid6ho;ficG cyclicho;ficG=vdjlI=l, aQ=1, aba-i=bmvamQ=1(modp) nhlfngm +1(modp) .Nellq khang1i1lfdcua (p-1)thitrltCfngh<;fpthri2khangxayra. 2.1.23.M~nhd'e.T 1anhomcap12dlt<;fCsinhbdi2ph'antztavab thoa: a6=1,b2=a3=(abj 22 .. 2.1.24.BpmIi. M6i nh6mG cap12khonggiaohoanho?ctilingdiu vdi A4ho#cD12ho#eT. 2.2.Mot s6thongtindinh11.1dngvanh6mSf . ,<' 2.2.1.Capeuanh6m55 I s51=5!=120=23.3.5 2.2.2.Caeudenguyendudngeua I 551 T~pta'tcacacu'ocnguyendu'dngcua I S511a: J={1,2,3,4,5,6,8,10,12,15,20,24,36,40,60,120} Theodinh11Lagrangene'uHsS5thl IHIE J 2.2.3.Caeph'antaeuaS5 Ca'utrucvongxich, s61u'Qng, ca'p)tinhchaD, Ie cuatungd?ngph'antli'cua S5du'Q'cth6ngke trongbangsail: 2.3.Mo tacaenh6mconGilaS5 D1favao[5J.va[11]d6motata'tcacacnh6mcon( ne'uc6) cuaS5lingvoi m6idEJ. Ca'utrUcvongrich S61uQng Ca'p T'mb ehan, Ie (i) 1 1 ChaD (ij) 10=(5x4): 2 2 Ie (ijk) 20=(5x4 x 3): 3 3 . chaD Ujkl) 30=(5x 4x 3x2): 4 4 Ie (ijkls) 24=(5'!):5 5 chaD (ij)(kl) 15= (5x4x3x2) 2 Ie2 2 2 (ij)(kls) 20=5x4x3x2xl 6 Ie ho?c(iik)(ls) 2 3 120=5! 23 2.3.1.d=l 55comQtnhomcandip l1a :Dl={Id} 2.3.2. d=2 55co25nhomcanca'p2 :10nhomcancodq.ngva15nhom concodq.ng £)~tD2i =={Id,(ij)} (i=1,10). D2j=={Id, (i j)(k I)} U =11,25) 2.3.2.1.10nh6mcand~ng: D2n=={Id,(ij)} (n=1,10) 2.3.2.2.15nh6mcond~ng: D2n=={Id,(ij)(k I)} D211= D216= D212= D217= D213= D21s= D214= D219= D215= D22o= 2.3.3. d=3 (n=1l;25) D221= D222= D223= D224= D225= GQim 1as6nhung3 nhomcanSylowcua55 D21= D26= D22= D27= D23= D2s= D24= D29= D25= D21O= 24 => { m=1(mod3) mt40 => m=1,4,10,40 NSum=40thlco itnhfttla 40x2=80ph'antli'cftp3khacnhau,maSs chico 20ph'antli'cffp3 lientacomQtmailthu~n,V~ym:::;10" Ta tIll thftySsco 10nhomconcffp3 cod(~mg: D3n=={Id,(i j k), (i k j) } (n=1;10) Giasli'H ::;ss, I H I =4, ta co H ==Z4ho~cH ==Z2XZ2 2.3.4.1 X6t trltC1nghQpH ==Z4 T~p{i,j, k, I}chotaSailchutrlnhcodQdai4,dola: (ij k 1),(i 1kj), (i kj 1),(i Ij k), (ij 1k), (i k Ij) VI ={Id,(i j k 1),(i k)O 1),(i 1k j)} = ={Id,(i k j 1),(i j)(k 1),(i 1j k)} = ={Id,(i j 1k), (i 1)0k), (i k 1j)} = lienling vdimQtt~pg6m4ph'antli'tachico3 nhomcancftp4d~ng cffuvdi Z4 D31= D36= D32= D37= D33= D3s= D34= D39= D3s= D31O= 2.3.4. d=4 25 ChQn4 phftntli'trong5phftntli'{i,j, k, 1,s},coCs4=5eachchQn vaellm6it~pcang6m4phftntli'cuat~p{i,j, k, 1,s}chota3nhom cand~ngca'uvoi Z4 Do do taco5x3=15nhomcand~ngca'uyoi Z4 Ssc615 nhomcand~ngca'uvoiZ4: 2.3.4.2.X6t trltCJnghqpH ==~xZz ([ 5,tr 13-15]) Ssco: * 15nhomcond~ngca'uyoi Zzx Zzcod(;l.ng D4n== {Id,(i j), (k 1),(i j)(k I)} =«i j),(i j)(k 1» (n=16;30) D416= D421= D4z6= D417= D4zz= . D4z7= D41s= D4z3= D4zs= D419= D4z4= D4z9= D4zo= D4zs= D43o= * 5nhomcond~ngca'uyoi~x~ co d(;l.ng D4n=={Id,(ij)(k l),(i k)(j l),(i 1)(jk)} (n =31,35) D431= D433= D432= D434= D435= D41= D4z= D43= D44= D45= D46= D47= D4s= D49= D41Q= D411= D412=«1354) > D413= D414= D41s= 26 2.3.5. d=5 GQiill las6nhung5nhomcon5ylowcua55 => { m::l(mOd5) mf24 =>ill =1 hay ill =6 =>m::;6 D5n=«i j k 1s»={Id,(ij k 1s),(ik sj l),(i1j sk),(is1kj)} V?y 55codung6nhomconcip 5.Do la: D51= D52= D53= D54= D55= 2.3.6. d=6 D56= Gia suH ::;55, I HI =6, ta co H ==Z6ho~cH ==53 . Xet tn1onghQpH ==Z6 55chico20ph'antuca'p6khacnhaucod~ng: (i j)(k 1s) Ma = ={Id,(i j) (k 1s),(ks1),(i j), (k1s), (i j)(k s1)}= V?y 55chico 10nhomconca'p6d&ngciu voiZ6 D61= D63= D65= D67= D69= . Xet tru'onghQpH ==53 D62= D64= D66= D6s= D61O= H co 10 d~ng : 5123,5124,5125,5134,5135,5145,5234,5235,5245,5345 Voi D6n=5ijk={rd,(i j), (i k), (j k), (i j k), (i k j) }=«ij),(i k» =«i j),(i j k» ( n =11,20) 27 2.3.7. d=8 Gia saH ~S5, I HI =8 GQim la s6nhting2 nh6mconSylowcuaS5 => { m==1(mod2) mIlS =>m=1,3,5, 15 =>m ~15 Ta Hmdu'qcl5nhomconca"p8 cuaS5. V?y S5co,dung15nhomcon ca"p8 ~':lng, trongdo : «i j),(i kj 1»=«i j k l),(ik»={Id,(ij),(ij)(k l),Uk),(il)Uk),(ij k 1), (i k)Ul),(i1j k)} D81= D82= D84= D8s= D83= D86= D87= D88= D89= D81O= D811= D812= D813= D814= D815= 2.3.8. d=10 GiasaH ~S5, I HI =10 Sur ra H ==ZlOho~cH ==DlO 2.3.8.1. TruonghdpH ==ZlO Ta tha"yH khongcoph'antuca"p10,nenH khongd~ngca"uyoiZlO 2.3.8.2. TruonghdpH ==DlO D1O=, , , 5 2 2=, , , E>~timH =trongd6s,tthuQcS5, I s I =5, I t I =2,(st)2=Id, 28 Ta coth~giiisus=(i j k 1Sl) (vI I s 1=5) ma khong ma-ttinhtang quat,r6ixettungtxemcos,tnaGthuQcv'eH mathGaI sI =5,I t I =2, (st)z=Id khong? I t I =2lienho~ct co d,;mgla mQtchuy~nvi ho~ct cod1;lngrichcila2 chuy~nvi dQcl?p. Dlfa VaGchungminhtrong [5], ta duQc: 55co 6nhomconca-p10d1;lng= {ld,(ij)(ks),(ik)(ls),(il)Uk), (is)Ul),Us)(kl),(ijkls), (iksjl) ,(iljsk),(islkj) } D101= D10z= D103= D105= D104= D106= 2.3.9. d=12 Giii suH ~55,I H 1=12 VI IH 1=12lienho~cH ~Z12,ho~cH ==Z6XZZ,ho~cH ==A4,ho~c H ~DIZ,ho~cH ==T voi T = a./ H ==ZlZ VI H khongcoph'antti'ca-p12lienH khongd&ngcf{uvoiZ12 b./H ==Z6xZZ ([ 5,tr21-23]) 55c6 10nhomconcf{p12d&ngcf{uvoi Z6XZZ c./ H ==A4.Ta c65nhomconH cila55,H ==A4,d6la : AIZ34, AIZ35, AIZ45, A1345, AZ345 Voi D 12n=Aijkl ={Id,(ijk),(ikj),(ijl),(ilj),Ukl),Ulk),(ij)(kl),(ik)UI),(il)Uk)} ( n =11,15) 29 d./ H ==D12Cacnh6mcondl;lngnaytrungvoi IOnh6mcondphanb./ e./ H ==TKh6ngc6nh6mconH naGcua55d~ngca'uvoi T V~y55c6 : 10nh6mconca'p12dl;lng: «i j )(k1s ),(k 1» = .=.=. ={Id,(kl),(ks),(1s),(ij),(ij)(ks),(ij)(ls),(ij)(kl),(ksl),(kls),(ij)(kls),(ij)(ksl)} (n =1,10) DI2t=. DI23=. DI2z=. D124=., DI2s=. DI27=. DI26= DI28=. DI29=.. DI21O=. 2.3.10. d=15 Ta tha'y55kh6ngconh6rncanca'p15.Th~tv~y,d@tha'yrnQinhorncan ca'p15d'euxiclic,matrong55kh6ngcoph'antti'naoc6ca'p15,dod655 kh6ngconhomconca'p15. 2.3.11. d=20 D1.!aVaG[7]d6chungmint B?t K= M=<ab I a5=1b4=1bab-1=a-z=a3>, " L=<ab I a5=1b4=1 bab-I=a-I=a4>, , , GQi H::;55, I H I =20 VI I H I =20,dod6ho?cH ==Z20, ho?cH ==K ho?cH ==L ho?c H ==M ho?c H ==Z2X22X Z5 ho?c H ==D2o al Xet trl1onghc;1pH ==Z20: 30 H~5s, H khongcophftntti'cftp20lienH khongd~ngcftuvdiZzo bl Xet tfl1C1ngh<;1pH ==ZzxZzXZs : Ta co ZzxZzXZs=ZzX ZlO Vi H khong co phftnt11cftp10lien H khongd~ngcftuvdi Zz xZz X Zs cl Xet tfl1C1ngh<;1pH ==Dzo: D ~I 10 1 z 1 -1zo=. Vi H khongCOphftntti'cftp10 lienH khongd~ngcftuvdiDzo d/Tn1C1ngh<;1pH ==Kho~cH==L ho~cH ==M Theo(2.1) , 55 CO30phftntti'cftp4 , do1a: (i j k l),(i j 1k),(i k 1j),(i k j l),(i lj k),(i 1kj)(i j k s),(ij s k),(i k sj),(i kj g), (i sj k), (i skj),(i j 1g),(i j s 1),(i 1sj), (i lj g),(i sj 1),(i s 1j),(ik 1s)(i k s1), (i 1k g),(i 1s k), (i s k 1),(i s 1k),Uk 1g),( j k s 1),(j 1k s),(j1s k) (j skI), (j s 1k) Vdi i,j , k, 1,s E {1,2,3,4, 5} Phftnt11atrongK, L, I a I =5 lientagias11a=(ij k 1s)ma khonglammftt lnht6ngquat Bay giotaxettungphftnt11b cocftpbang4 d~xemcotru'ong hmaH ::K ho~cH ::M khong? . Ne'ub=(i j k 1) a.b=(i j k 1s).(ij k 1)=(i k s)(j1)~H . (vI I H I =20,khongth~chuaphftnt11cftp6), talo~ib =(i j k 1) VI (i jk 1s)=(j k 1s i) =(k 1s i j) =(l s i j k) =(s i j k 1) (**) lien taloc;Udu'<;1cthem4 phftnt11'cftp4 . 80 1a : U k 1g), (k 1s i) =(i k 1g),(l s i j) =(ij 1g),(s i j k) = (i j k g). . Ne'ub=(ilkj) ={Id,(i 1kj), (i k)(lj),(ij k I)} 31 (1j k 1sleij k I) =(i k s)UI) ~H (do IH I=20 lienH khongth~chuaph'antti'ca'p6 )talo<;lib=(i1kj) Do (**) lientalo<;lithem4 ph'antti'm1'al : Us 1k), (k is I) =(i s1k), (lj i s),=(i sIj), (skj i) =(i skj) V~ytdngcQngtalo<;li10ph'antti'ca'p4. . N€u b=(i j 1k) Ta tha'ya2=(ij k 1s)(ij k 1s)=(i k sj I) a'-l=(s 1k j i) =(i s1kj) bab-1=(ij 1k)(ij k 1s)(kIj i) =(i k sj 1)=a2 khongdAngca'uvoi L VI bab-1:;t:a-l = {rd,(i j 1k), (i I)U k)(i k 1j)} , d~dangtha'y=<a,(i k Ij» Ta co(i k Ij)(ij k 1s)U1k i) =(i Ij sk)=a-2,(i Ij sk) :;ta2,a-l Suyra<a,(i k 1j» khongdAngca'uvoi K tucla khongdAngca'uvoi K nhu'ng==M lien tachQnb=(ij 1k) Do (**) lientachQnthem4ph'antti'nuala : U k s1),(k1is) =(i ski), (l sj i) =(i 1sj), (&i k j) =(i k j s) . N6u b=(i k 1j) bab-1=(i k Ij)(ij k 1s)U1k i) =(i Ij sk) =a-2 suyrakhongdAngca'uvoi K,L , nhung==M Theo (**) tachQnthemdu'Qc4ph'antti'nuala : U 1sk), (k s i 1)=(i 1k s), (1i j s)=(i j s1),(sj k i) =(i sj k) . N6u b =(ikj 1) bab-1=(ikj 1)(ijk1s)(ljk i)=(iskI) :;ta2,a-I,a-2lien lo<;lib. Theo (**) talo<;lithem4 ph'antti'nuala : U1k s),(k s1i) =(i k s1),(l i sj) =(i sj 1),(sj i k) =(i k sj) 32 . Ne'ub=(i 1j k) bab-I=(i 1jk)(ijk 1s)(kj1i)=(ij s1k)"*az,a-I,a-zlien1o~ib. Thea(**).ta1o';lithem4ph'antli'nua1a: (j sk 1),(ki 1s)=(i 1sk ),(1j s i)=(i 1j s ),(sk i j )=(ij sk) Ta thfty: C6 20 ph'antti'cftp4 d~ubi lo';lilienthong c6 nh6mcan H ==K ho~cH ==L. V~y: 55 c6 6 nh6mcancftp20 d~ng: D20n=«ijkls ),(jk~l» (n =1,6) D201=«12345),(2354» D203=«12435),(2354» D20z=«12354),(2345» D204=«12453),(2435» D205=«13245),(2345» D206=«13452),(2435» 2.3.12.d=24 [ 5 , tr28-29] C65nh6mcancftp24: 5ijkl= D241=51234 = D242=51235 = D243=51245 = D244=51345 = D245=5Z345 = 2.3.13. d=30 [ 5 , tr29-30] 55khongc6nh6mcancftp30 2.3.14. d=40 [5 , tr30-33] 55khongc6nh6mcancftp40 2.3.15. d=60 [ 5, tr33] 55 c6 1nhomcancftp60la A5 33 As = {Id ,(123),(124),(125),(134), (135), (145), (132), (142), (152),(143) ,(153),(154),(234),(235),(345),(243) ,(253), (354),(245),(254),(12)(34),(12)(35),(12)(45),(13)(24),(13)(25), (13)(45),(14)(23),(14)(25),(14)(35),(15)(23),(15)(24),(15)(34), (23)(45),(24)(35),(34)(25),(12345),(13524),(14253),(15432),(12354), (13425),(15243),(14532),(12435),(14523),(13254),(15342),(12453), (14325),(15234),(13542),(13245),(12534),(14352),(15423),(13452), (14235),(15324),(12543)} =«123),(124),(125»=«12)(34),(25)(34),(13)(24». 2.3.16. d=120 5s={Id, (12),(13),(14), (15),(23),(24),(25),(34),(35), (45),(123), (124),(125), (134), (135), (145), (132), (142),(152), (143), (153), (154), (234), (235), (345), (243), (253), (354), (245), (254),(12)(34),(12)(35), (12)(45), (13)(24),(13)(25), (13)(45), (14)(23), (14)(25), (14)(35), (15)(23), (15)(24), (15)(34), (23)(45), (24)(35), (34)(25), (1234), (1235), (1245), (1324), (1325), (1345), (1432), (1532), (1542), (1423), (1523), (1543), (1243), (1253), (1254),(1342), (1352), (1354), (2345), (2453), (1453),(2543),(2534), (2435),(1452),(2354),(12345),(13524),(14253),(15432),(12354), (13425),(15243),(14532),(12435),(14523),(13254),(15342),(12453), (14325):(15234),(13542),(13245),(12534),(14352),(15423),(13452), (14235),(15324),(12543),(13)(245),(13)(254),(14)(235),(14)(253), 34 (15)(234),(15)(243),(23)(145),(23)(154),(24)(135),(24)(153), (34)(125),(34)(152),(35)(124),(35)(142),(45)(123),(45)(132), (12)(345),(12)(354),(25)(143),(25)(134)}= Vi~cmatacacnh6mcanD2o,D24,As,5stadungcacl~nhtrongphftn m~mMaple [ Ph\llt;lc1] 2.4. Vai diicdidmCUBcacnh6mconCUBnh6mSf=-G . DvavaomQts6ly thuy€tc6lienquandiigidithi~ud2.1,vi~cmata cacnh6mcancua5svatucangthuctrongdinhly2.1.7: c=IHgl=[G:No(H)]=IGI:'INo(H)1suyra INo(H)1=IGI: c (2.4.*) lu~nvansephanlo<;ticacnh6mcancua5svaxacdinhchu§'nboaill'cua chung 2.4.1.511lienh<;1pcuacacph'anor, caenh6mcantrong5s Cacphftntuc6clingca'pva clingca'utrucvongxfchcua5sd~ulienh<;1p voi nhau( dom~nhd~2.1.17).Tli d6suyracacnh6mcanclingca'pva cunglo<;ti.d~ulien h<;1pvoinhau. Vi d\l : 35nh6mcanca'p4 chialam3lo<;ti: * 15nh6mcancyclicd<;tng«ijkl» ==Z4 lienh<;1pnhau,VI voi a =(ijkl)va ~=(jksl)taluauHmdu'<;1cy=(ijks)E 5ssaochorail =~ * 15nh6mcand<;tng«ij),(kl» ==Z2XZ2lienh<;1pvoinhau * 5 nh6mcand<;tng«ij)(kl),(ik)(jl» ==Z2XZ2lienh<;1pnhau 2.4.2.Phanlo~icacnh6meoneuaSs 2.4.2.1.Cacnh6mcanca'p2 : C6 25nh6mcanca'p2 du'<;1cchialam2lo<;ti 35 . 10nh6mcancyclicH ca'p2d,;mg«ij» Ta c6 I NG(H)I=120:10=12lien chu§:nhoatli'cuam6inhomcanca'p2 «ij» lamQtnhomcanca'p12(dihedral)«ij)(kls),(kl» . 15nhomcanca'p2cyclicH dC;lng«ij)(kl» Ta co I NG(H)I =120:15=8, nenchu§:nh6atli'cuam6inh6mcanca'p2 naylamQtnh6mcanca'p8 «ij)/ikjl» ~Ds 2.4.2.2.Caenh6mcanea'p3 Co 10nh6mcancyclicH ca'p3dC;lng«ijk»lien hQpnhauva I NG(H)I =120: 10=12, nenchu§:nhoatli'cuam6inhomconca'p3 la mQtnhomcan ca'p12dC;lngdihedral( theom~nhd'e2.1.15) M6i nhomcanca'p3la 3-nh6mcanSylowcuaS5vala nh6mcantoidC;li cuaS5 2.4.2.3. Caenh6mcanea"p4 : Dtfavaod~ngthuc(2.4.*) taxacdinhduQcchu§:nhoatli'cuam6inh6m canca'p4 dC;lng: . «ijkl» la mQtnhomcanca'p8dC;lng«ijkl),(ik» . «ij)(kl» la mQtnhomconca'p8d,;mg«ijkl),(ik» . «ij)(kl),(ik)(jl»la nh6mcan ca'p24 d~lllgSijkl 2.4.2.4.Caenh6mcanca'p5 : C6 6 nhomcanH ca'p5 dC;lng«ijkls» lienhQpnhauva ING(H)I =120:6= 20 , lienchu§:nhoatli'cuam6inh6mcanca'p5 la mQtnhomconca'p20 dC;lng«ijkls),(jksl» , m6inhomconca'p5 la mQt5-nhomcanSylowcua S5,nenn6la nhomcantoid?i . 2.4.2.5.Caenh6mcanea'p6 36 C6 20nh6mconea'p6chia lam2 lo~i, caenh6mconeuam6ilo~i lien h<;1pnhau . 10nh6mconcyclicH ea'p6 d~ng«ij)(kls» , ehuftnhoarii'eua m6inh6mconH ea'p6naylamQtnhomconea'p12d~ngdihedral «ij)(kls),(kl» VI I NG(H)I =120:10=12 . 10nhomconH ca'p6 d~ngSijk==S3, ehuftnhoatli'cuam6inh6m . connaylamQtnh6mconea'p12d~ngdihedral«ijk)(ls),(ij» 2.4.2.6. Caenh6mconea'p8 C6 15nh6mcon ea'p8 d~ng«ij),(ikjl» , chuftnh6a tU'cua m6i nh6mcon H ea'p8 la mQtnh6mconea'p8 vIING(H)I= 120;15=8, tueIa chungtvehuftnh6a.M6i nh6mconca'p8la 2-nh6mconSylowcuaS5,la p- nh6mcont6id~icuaS5 2.4.2.7.Caenh6mconea"p10 C6 6 nh6mconea'p10d~ng«ijkls),(ij)(ks» ==DlO, lienh<;1p~au) VI ING(H)\=120:6=20Denehuftnh6atU'cuam6inh6mcanH ca'p10la mQtnh6mcanea'p20 2.4.2.8.Caenh6mconca"p12 C6 15nh6mconca'p12chialam2lo~i: . 10nh6mconea'p12d~ngdihedral«ij)(kls),(kl» , m6inh6m canH naytvchuftnh6avIING(H)I=120:10= 12. Cancuvaoph'antii'sinh tatha'ym6inh6mconea'p24khongchuaph'antU'ea'p6va A5khongehua caeehuy~nvLnenm6inh6mcannayt6id~itrongS5 . 5 nh6mcan ca'p12d~ngAijkl==~ , maA4 <JS~=SijklDenm6i nh6mcanea'p12d~ngAijkle6ehuftnh6atU'lamQtnh6mcanea'p24. 37 2.4.2.9. Caenh6mcanea'p20 Co 6 nh6rncancffp20 d~ng«ijkls),Uksl» , rn6inh6mcanH nayt1;( ehugnhoavIING(H)1=120: 6=20vachungt6id~i( VIAskh6ngehuaph'antii effp4 ) 2.4.2.10.Caenh6mcanea'p24 C65nh6rncancffp24d~ngSijkl==S4, chungtvchugnh6aVI INQ(H)I=120:5=24 .Ta c624kh6ngla tidecua60nenchungt6id~i 2.4.3. Quanh~"baaham" giuacaenh6mcancuaSs I . As chua:- Cacnh6mcaneffp12( d~ngAijkl) , cae nh6rneon:effp10j car5-. - Caenhorncaneffp3(Aijk),eaenh6rnconeffp4( d~ng «ij)(kl),(ik)Ul» ) - Caenh6mcancffp2(d~ng«ij)(kl») . M6i nhorn can effp24 ( Sijkl)chua : - 3nhorncaneffp8 :«ijkl),(ik»,«ikjl),(ij»,«ijlk),(il» - 4 nh6mcaneffp6 :Sijk,Sijj,Sjkl, Sikl 1nhomcaneffp12:Aijkl - 7nhomcaneffp4( 3nhomcaneffp4lo~i1,3nhorncaneffp4 lo~i2, Inhorncaneffp4lo~i3) - 6nhomcaneffp2d~ng«ij»,«ik»,«il»,<Uk»,<Ul»,«kl» - 3nhomcaneffp2«ij)(kl»,«(ik)Ul»,(il)Uk» - 4nhomcaneffp3 :«ijk»,«ijl»,<Ukl»,«ikl» M6i nhornconeffp20ehuadung1nhorncanea'p10, 1nh6mcaneffp5. , 5 nh6rncaneffp4 cyclic, 5 nh6mcaneffp2 d~ngtich2 chuySnvi «ij)(kl» . M6inh6rncandip 12d~ng: 38 - Dihedral «ijk)(ls),(ij» chua2 nhomconca'p6 : «ijk)(ls»,Sijk , 1nhomconca'p3 ( Aijk), 3nhomconca'p4 d~ng «ij),(kl», «ij),(ls», «ij),(ks» - AijkIchua dung4 nhomcon ca'p3 ( Aijk ,Aiji ,Aikl,AjkI) . M6i nhomconca'p10:chuadung1nhomconca'p5 . M6i nhomconca'p6 : . - D~ngcyclic«ijk)(ls» chuadungmN nhomconca'p3 (AijJ, 1nhomcon ca'p2 «ls» - D~ngd~ngca'uvoi S3:duQcchuatrong2 nhomconca'p24la Sijkl , Sijks . M6i nhomconca'p8 «ij)(ikjl» chua: 3nhomconca'p4lo~i1, 3nhomconca'p4 lo~i2,1nhomcon ca'p4 lo~i3) - 2 nhomconca'p2 «ij»,«kl» 3 nhomconca'p2 «ij)(kl»,«ik)UI»,«il)Uk» . M6i nhomconca'p4 : - D~ng«ijkl» duQchuatrong2nhomcolica'p20, 1nhomcon ca'p8vanochua1nhomconca'p2la «ij)(kl» - D~ng«ij),(kl» duQc huatrong1nhomconca'p8 vanochua1 nhomconca'p2 «ij)(kl», 2 nhomconca'p2 <ij»,«kl» - D~ng«ij)(kl),(ik)(jl» chua3 nhomconca'p2 d~ngthu3 , no duQcchuatrong3nh6mconca'p8 . . M6i nhomconca'p3 «ijk» duQcchuatrong2 nhomconca'p12: Aijkl, AijksvachuatrongmQtnhomconca'p12 dihedraltudngling. 39 M6i nhomconca'p3 la 3-nh6mconSylowchu§:nt~ctrongnh6m conca'p12dihedral(theom~nhd~2.1.15) . M6inh6mconca'p2 : - D(~mg«ij» lanh6mconchu§:nt~ccua3nhomconca'p4 : «ij),(kl»,«ij)(ks», «ij),(ls» , tud6nodu'<;1cchliatrong3 nh6mconca'p8tu'dngling. dod6, du'<;1cchli'atrong3 nh6mcon ca'p24 : Sijkl,Sijks, Sijlsva n6 dU<;1cchlia trongmQtnhomcon ca'p 6 cyclic: - D,;mg«ij)(kl» la nh6mconchu§:nt~ccua3 nh6mconca'p4 ( m6iloq.i1nh6mcon)«ikjl»,«ij),(kl»,«ij)(kl),(ik)(jl» * M6i quailh~baohamgiUamQts6nh6mcon d<;lidi~ncua55du<;1c minhhQatrongphl,lll,lc4 * Bangphanlo<;licacnh6mconvachu§:nh6atlrcuachungduQcth6ng ke trongphl,lll,lc3 .