Luận văn Sự sắp xếp các nhóm con của nhóm tuyến tính GL (2,Q) chứa xuyến không chẻ

§12.C~p DOl Trang phftnnay ta giii thie'tA la modulch~pnh~ndtI'</ctrongQ VOlv~mh nhanrii'R va H la nh6mcantrunggianVOlA(H) =A. Trang §7, VOlmai nh6m * . cantIlinggianH, tadac61\(H)Ia nh6mcancuanh6mR , d day, L\(H)= {8EQ' :(~ ~)E H} Ne'u1\la nh6IDcan biltky c_uanh6mR* thlkh6ngh~nse t6n t<:1iID<)tnh6m cantIlinggianH saochoA(H) =A va1\(H)=1\.TITd6 tac6d!nhnghlasail. 12.1Dinhnghia: * Modul A trongQ vanh6mcan1\trongQ du</cg9i la m<)tcgpdol, hayL1c(ip doi veiiA, ne'ut6nt<:1iID<)tnh6mcantrungH saochoA(H) =A va1\(H)=1\. Taduavaakyhi~u no(A) ={8E R*, 82- 1 E A} D@thilyOo(A) la m<)tnh6mcancuanh6mR*. TITBinh 1:98.4tasuyra tn;(ctie'pcacb6d~sail. 12.2Blf de : Giii sad chiin.Khi d6 Oo(A)= { 8ER*: 2d (82_1)EA,'\IXE Q}x -d = { 8E R*: x (82-1)E A, '\IxE Q } . x2 -d 12.3Bd de : Giii sad chiin.Khi d6 ° * O(A)={8ER :g(X,8)E(l+A)*,'\IXEQ}. Changminh VOl ID9ix E Q,8 E Oo(A) rheac6ngthil'c7 §3 tac6 53 g(x,e)=1+ d e2-12 .- X - d e2 =1+~. d fe2-1). 82 x2 -d ~ DoeE R* nen, 12 =e-2 ER*, vatheoB6de12.2thl d (82-l)E A.e x2-d * V~yg(x,e) E (1+A) . 0 * 12.4Dinh If : Gia sad chanvaA Ia nh6mconcuanh6mR . Khi d6cackh£ng dinhsauIatu'dngdudng: 1)~ci;ipdoivdiA va~::;;Oo(A). 2)H =T( ~ ~}anh6mcon!rungiand~ng1. 3)~thoacacdieuki~n: a) 00 (A) ::;;~::;;no(A); b) ~baa hoa, nghla Ia g(X[NKHP1],0)E ~, Vx E Q,o E /).. Chang minh 1) .::::>3) Gia sa ~ ci;ipdoi vdi A va ~::;;Oo(A). Theo Dinh nghla 12.1thl t5n t~imQtnh6mcontrunggianH saochoA(H)=A; ~(H)=~.Khi d6 Fo(A) ::;;H va ~(Fo(~»=00 (A) (theoDiM ly 11.1).Suyra00 (A) ::;;~,kh!ngdinha)duQcchung minh.Giasa0Iaph~ntii'b~tkytrong~.TheoH~qua3.3.thlvdimQix E Q tac6 (~ ~)G :) ={~~), t E T, (1) Trongdo p=!. X (02-1), Y =og(x,o).Do 0 E ~ ::;;Oo(A) nen 0 x2-d 0 E Oo(A), theo B6 d~ 12.2 thl x (02-1) E A, suyra p E A.Mi;it khac, x2 -d 54 matr~nG ~)E H; snYfay=og(x,o) E 11..Do 0 E 11.n6n g(x,0) E 11..Khing dinh b) duQcchungminh. ( 1 O J . 3) =?2)Giii sa3)duQCfuGa,tachungminhH =T A 6 Ia mQtnh6m.Do ( 1 0 ) ( 1 0 J ( 1 0 ) ( 1 O J ,? H =T =T =Fo(A) nenta chi din kiem ITa A 6 A no(A) o ~ o ~ ( 1 0 ) ( 1 o J . nhom hoanvidudcv6inhom .Giiisa8 E ~batky' , vax E Q. o ~ .. A no(A) Do 3)duQcthGamannen8 E no(A) vag(x,8)E A Voi x va8 nhutrenthltac6 matr~n(~ ~J trong(1);vOiJ3E A (theoB6M 12.2vado/; E [:!o(A»vaY E '" (dotinhbaoboacuanh6mcoo). V~ymatr~o(~ ~) E H. soyfa nh6m(~ ~) hoanvi du'<Jcv(ji nMm Fo(A). Ta di Mn, H = Fo(A{~ ~) Iii nh6m va hi1!n nhienH lanhomcond~ng1. 2)=:>1) Gi:l SIT H =T(~ ~) Iii nh6mcon !runggian d~ngl. Khi d6 A(H) =A, ~(H)=!1nen~ c~pd6i voi A. Hdn nii'adoH Ia nh6mcontrunggiand<;tng1 nenA(H) =B(H) =B. TheoBjIlh ly 7.11taco r2::;;(1+2B)*, ma~::;;r Suyfa 2 * * ~ ::;;(l +2B) = (l +2A) , vakhid6~::;;no(A).0 12.5Dinh Ii : Giii sad ch[nva A Ia modulchapnh~nduQCtrongQ. Khi d6 nhomconQo(A)c~pd6ivoiA. 55 Chang minh Hi€n nhien ta c6 00 (A) ::::; Oo(A).B€ chungminhOo(A)c~pdoi vOiA tad.n chungminh H =T(~ QO~A)J Ianh6mcontrunggiand~ng1. ( 1 0 ) ( 1 0 ) ( 1 0 J Tac6H =T =F A A °o(A) ° no (A) o() ° 0° (A) B€ chungminhH Ia nh6mconthl tacftnchungminhnh6m ( 1 0 0 J0 Q (A) hmlnvi vdi nh6mFo(A).R6 rang, ( 1 0° ) hminvi vOi ( 1 o J . Giasa . o 0 (A) . A 00(A) 0Iaph~ntti'ba'tky trongOo(A) vax EQ.TheoH~qua3.3taco ( 1 O J ( X d ) - ( 1 O J - t , t ET, 00 Ix po vdiP=~2x (02-1), Y=og(x,o).Ox -d Do 0 E OO(A)DentheoB6 d~12.2ta co pEA va rheaB6 d~12.3ta c6 g(x,o) E (l + A) *, suy fa g(x,o)Z- 1 E A, nghla la g(x,o) E no(A). Ta di dtn ° ( 1 O J - , y =og(x,o) E 0 (A), khi d6matr~n p y E H. V~yH Ia nh6mconvadethfty H lanh6mcond~ng1. * 12.6B6 d~: Giasadch[n,x E Q va0 E (1+dA) . Khi d6g(x,o)E O(d,A). Changminh * Da0E (1+dA) nen0c6d~ng8=1+da,a E A.Theachungminh7§4tac6: g(x,o)=uZ- dvz, 56 ~. 1 d 1 d x 1 dVOIU= + .-. a;v= .-. a. x2-d 8 x2-d 8 TheoBinh ly 8.4taco cacs6 d , x chuatrongR. Ta di de'nU ==1 . x2-d x2-d (moddA), v ==0 (modA). TheoBinh nghla(j §12 tacog(x,8)E Q(d,A). 12.7Menhde : Giil sadchan.Khi donhomQ(d,A)c~pd6ivoiA. Chungmink Giilsa81aph~ntii'b~tkycuaQ(d,A),khido8=u2- dv2voiU==1(moddA), v==0 (modA). Suyfa 2 2 8=(1 +da) - d~ voia, ~E A. Ta di de'n8 E (1 +dA)*, theoB6 d~12.6taco g(x,8) E Q(d,A). M~tkhac taco no (A) :::;Q(d,A) :::;Qo(A),nenrheaBinh ly 12.4tac6Q(d,A) c~pd6i vdi A. 0 * 12.8Dinh If : Giil sa~t.~2:::;R c~pd6ivoi A. Khi d6~1.~2va~l (\ ~2cling c~pd6ivoiA. Chungminh Do L11,L12c~pd6i voi A nent6nt~iHI, H2 saocho~l(Hr) =L11,L12(H2)=~2, A(HI) =A(H2) =A. B~ chungminh ~l (\ ~2c~pd6i vdi A ta c:ln chungminh t6n t~inh6mcon trunggian H saocho ~l (\ ~2(H)=~l (\ ~2,A(H) =A. Ta chqnH =HI (\ H2,khi d6 ~l (\ ~2(H)=~l (\ ~2vaA(H)=A. Tie'prheatachungminh~1.~2c~pd6ivoiA. Giil sa81E ~l, 82E ~2, \/x E Q theoc6ngthuc10§3 tac6 g(X,8182)=g(x,81).g(8IX,82). Do ~l c~pd6i vdi ~ nen rhea Binh 1:912.4 ta co g(x,8d E ~l, tu"dngttf 57 g(OlX,OZ)E dZ'Do d6g(x,o18z)E dl.dZ' V~ydl.dz c~pd6i voi A. 0 12.9Dinh If: Neu d=:;Qo(A)c~pd6ivoi A thlnh6m III * dell) ={8Ed: 8 E (1+dA) }, m~O c~pd6ivOiA. Changminh * Do no (A) =:;(l + dA) nen no (A) =:;ll(m), suy fa 00 (A) =:;d(m) =:;Qo(A). Nhu' v~y,d~chungminhll(m)c~pd6ivoiA thltheoDiM 1y12.4tachIdin chUngminh * voi0 E dell), x E Q thlg(x,8)E ll(m),di~unaytu'ongdliongvoig(X,8)ffiE (1+dA) . Taco g(X,O)ID=1+((o2g(x,o)f -l)+(l-OID )~+8m)g(x,o)m. (1) Do8 E ll(m)nen1- 8IllE dA, VI the's6h,;lllgcu6iclingcuading thuc(1) thu9cvao dAvatacftnchungminh(82g(x,8)f-1 E dA.Theoc6ngthuc7 §3tac6 2 02x2-d 8 g(x,8)= 2 .x -d Ta di de'n, - (82x2df _(x2-d)m. (o2g(x,o)t-1- (x2-dr Tuynhien m-k (82x2 -df =82mx2m +L(-ltC~8Z(m-k)x2(m-k)dk +(-I)mdm , k=l m-k (x2-df =x2m+L(-I)kC~x2(m-k)dk +(-l)mdm, k=l 58 Suyfa (02g(X,0))m-1 = ( ;2 J m(om+1XOm-1)+X -d +~(-l)kC~ ( ;2 J m-k-1 ( 2X J 2 ( 2d J k-1 d(02(m-k) -1.L..J X -d X -d x-d k=l 2 Th D' h 1/ 84 / / ,.:" X X d th ~ , R d d 'eo LJ~n y . ta co cacso 2 ' 2 ' 2 UQc vao , 0 0- X -d X -d x-d (02g(x,0)r -1 E dA. V~y tUd~ngthuc(1) ta co g(x,o)mE (1 + dA)*. Dinh 19 du'<;1cchungminh. 0 g(x,a2") 12.10B6 d~: Ne'u8 E Qo(A)thivdim6iso'X E Q taco on= , n ~ 1, g(x,8)2n chuatrongfleA). Changmink Ta chungminhb6 d~b~ngquin<;1ptheoll. Vdi n =1,theocongthuc7 §3taco ( 2 J 2 ( 2 J 2 2 d 8 -1 X 8-1 g(x,8 )= 1+ 2 . 2 - d 2 . 2X -d 8 x -d 8 =g(X,8)2 [ 1-d ( 1 . X .82_1 J 2 } g(x,8) x2 -d 82 . Ta di de'n ( ,,2\ 2 01=~=1-dv2 , v= 1 . X .8 -1. g(x,8)2 g(x,8) x2 -d 82 TheoB6 d~12.2tacov E A (do8 E Qo(A)),Den 01=1- dv2E (1- dA2);ma 59 1- dA2c Do(A) (theoB6 d~7.1vaBinh 11.1),suyfa 01E no (A) vadod6 01E n(A). B6d~dungvain =1. Giasab6d~dungvOitntonghQpn- 1(n2::2),nghlaIatac6 ( n-1 ) g x,82 °n-l = . n-l E D(A) g(x,8)2 n-1 Do 8 E no(A) Den 82 E no(A), ta c6 g(x,OZ") ( 2n-1 ) 2 E n(A) g x,8 Suyfa, 0 =g(X,92) =O~-l g(X':n~)2 E Q(A) n g(x,a)2 g(X,a ) B6d~dungvoinvab6d~duQc hungminh.0 12.11Dinh1i1GiasaLl~no(A)c~Pd6ivoiA. Khid6tac6cackhiingdiM sail: 1)Nh6mcon Ll2nno(A), n 2::0,c~Pd6i voi A. 2) Ne'ud chilDthinh6mcon ti(0) ={eE Q 0(A) :e2nE A}, 0 " 0,c~puBivOiA Chungminh 1) Ta chI cfin chungminhne'uLl ~ no(A) c~pd6i voi A thi nh6mcon Ll2.no(A)c~pd6ivoi A. TheoBinh1)'12.4tachIcfinchungminhtinhbaohoa cuanh6mconLlZ.no(A).Gia sa8 E Ll2.no(A),s68 dl1qcvie'tdl1oid~ng8?.8z, 60 vOi81E ~,82E no(A). Voi m5ix E Q theocongthuc10§3 taco: 2 2 2 g(x,8) =g(x,81.82)=g(x,81).g(x81,82)' Do 81E ~c~pd6ivoi A lientheoBinh ly 12.4tacog(X,81)E ~.TheoB6 d~12.11 tacog(x,el) E fl2.00(A).Tttongt1!g(8lx,82) E DCA).V~yg(x,8)E ~2.n(A).Hon Qua,ne'udchiin ho~cV2(A)=1=2 thl theoH~qua 11.2tac6 DCA)=no(A). V~y g(x,8)E ~2.no(A),suyfa ~2.no(A)baaboa,rheaBinh ly 12.4tac6~2.no(A)c~p doivoiA. 2) Chungminhtu'ongtl!nhtt1).D 12.12B6 d~: Gia sa d chiinho~cV2(A)=1=2. Khi d6 c6 mQttu'ongling 1-1 giUacac nh6m conLl~no(A)c~pd6ivoiA vachIsO'(no(A): ~)voicacnh6mcon~1c~pd6i * * * vdi A thoa (1 +dA) ~Lll ~(1 +A) va ((1 + A) : Ll})Ie. Changminh B~t f}J={~: ~~no(A), ~c~pd6ivoi A, (no(A) : ~)Ie} * * * f}J1={~1: (1 +dA) ~~1~(1+A) , ~1c~pd6ivoi A, ((1 +A) : ~1)Ie} va taxacdinhcacanhx~11,111nhttsan: 11:f}J ,f}J1 111: f}J1 , f}J * ~~~n(1 +A) ~1~ ~1(1) vdi Li1(1)={8E no(A) : 82 E ~d Ta ki€m rIa cacanhxa11,111du<;1cdinhnghIato't.Trudc tien taki€m rIa anh x~11. 61 * *2 GiiiSa~ E $. Ta co (1+dA) ~OO(A)ma (1+dA) ~Oo(A)(theoB6 d~ 11.3)nen(1+dA)* ~~(doOo(A)~~va(OO(A):~)Ie).Ta diden(1+dA)* ~ * * * ~n (1 +A) ~(1 +A) .M~tkhactheoBinhIy 12.8thl~n (1.+A) c~pd6ivoi A. KIStd6ngcftu: p: (1+A)* ~ OO(A)/~ Taco Kefp= (I +A)* n~nen (I+A)*/(1+A)* n~~OO(A)/~ - * * Do (Oo(A): ~)Ie Den((1+A) : (1+A) n~) Ie.V~yanhX(;l11du<jcxacdinh tat. A.nhX(;l11 du<jckiem ITatu'clngtt!. Tiep theotachungminh11=11 1.Giii sa~E $, ~1E fih taco: * 11111(~)=111(~n (1 +A) ) 2 *={8E no(A) : 8 E ~n (1+A) } ={8E OO(A): 82 E~} (Do 8 E Oo(A) thl 82 E (1 + A)*) ={8E Oo(A): 8 E~} (Do (OO(A):~) Ie} =Ll 11111(~1)=11(~1(1» ~ * =~1(I)n(1 +A) * 2={8E (1+A) : 8 E ~d * * ={8E (1 +A) : 8 E ~d (Do ((1+A) : ~1)Ie) =~1' Suy fa 11=1111.B6d~du<jchungminh.0 62 */ * *TheaBinhly 6.5va6.6tac6 R (1+aR) ~ (ZjaZ) vadad6nh6mthu'ong */ *R (1+aR) c6ca'pla <pea),vdiA =aR va <pla hamEuler.TiI ke'tquanayta tha'yf~ngvdiso'bnguyenc6d~ngb=q~I...q~n,trangd6qi ~R*, ki 20 voi 1~i ~n, thinh6mthuong(1+A)*/(1+bA)* c6ca'pla <p(ab)<p(afl voi A =aR. 12.13B6 d~: Gia sah~ng6ndinhcuavanhR b~ng1,A :j;(0)la ideancuaR. Ne'u p la s6 nguyen to' khong kha nghich trangR thi nh6m thu'ong (1+A)* /(1+pA)* la nh6mxiclicc6 ca'pla p ne'up I a va c6 ca'pla p - 1 ne'u p % a,vdi A =aR. Changminh TiI nh~nxettrentatha'yca'pcuanh6m(1+A)*/(1+pA)* b~ng<p(pa)<p(a)-I. M~tkhac <p(pa)'<pea)-1 = { p p-l ne'up a ne'up%a Ke'tie'ptachungminhnh6m(1+A)*/(1 +pA)* la nh6mxiclic.Ne'up I a thi */ *rheatrentac6ca'pcuanh6m(1+A) (1+pA) b~ngp Denn6la nh6mxiclic. Ne'uP% ataxetd6ngca'utl!nhien <Pp: (1+A)*~ (R/pR)* * * * HitSnnhien Kef <Pp=(1+A) n (1+pR) =(1+pA) . Suyfa (1+A) */(1+pA)* ~ (R/pR)* M~t khac nh6m xiclic (Z/pZ)* c6 ca'pla <pep)=p - 1 b~ngvoi ca'pcua (1+A)*/(1 +pA)* .Tadide'n */ * ( )* *(1 + A) (1+ pA) ~ R/pR ~ (ZjpZ) . 63 V~y (1+A) */(1+pA)* la nh6mxic1ic.0 Gia sitA =aR la ideancuaR, vdia la phfintii'sinhcuaA. Gia sitd la mQt s6 nguyenkhong c6 u'dcblnh phuong.Ta phftn tich so d du'di d~ng * .* d =dO.PI ... Pc.Pc+I ... Pc+s, voi doE Z n R ; I, S ;:::0; Pi \itR , VI :::;i :::;I + s; . It \-11<.< . . 1 \-I 1<.< . 1:"\- d - . H'~ PI ~ a, v - 1 - I, PI a, vI + - 1- I +s. L1~t I - PI... Pc.Pc+1... Pc+s. ten nhien,d=dad!vadR=diR. 12.14Menhd~: Gia sith~ngan-dinhcuavanhR b~ng1,A * (0) Ia ideancua R, d la mQtso nguyenkh6ngc6 u'dcblnh phuong.Khi d6 nh6mthuong (A)= (I+A)*/(1+dA)*c6 dip la cp(dIa).cp(a)-Iva duQcphftntich thanh d~ngtich (A)=CI ...Cr+s~CJ x... X Cc+s' y6i OJ =(1+:i Ar)1+dA)' . 1:;; i :;; r +s.Hon nffaCi1. nh6mxiclic c:!pPi - 1 neu 1 :::;i :::;I va c6 dip Pi neu I + 1 :::;i :::;I + s. Changminh Cac kh~ngdinhcuoiclingcuam~nhd~du'QcsuyIa tUB6 d~12.13. Voi 1 :::;i :::;I + s ta d~t: Ci =I1cj j;ei HiGn nhien Ci :::;(l+PiA)*/(l+dA)* * } - d * * ci nci :;:;(1+Pi A J n(l+piA) (l+dA) Ta di den 64 *Ma (1+~AJ n (1+. Pi A)* =(1+dA)*, Suyfa Cin Ci Ia ddn vi cua nh6mPI . thu'dng<D(A).Suyfa tich @=Cl"'Cr+s la tichtn1ctie-p,nghlaIa @ ~ Cl x ...x. Cr+s. Ta di de'n r r+s 1 cafd@ =II (Pi -1) fIpj =<p(d,a).<p(a)- . i=1 j=r+1 * * M~tkhacdod =dodl,vdidoe R*, Den(1+dA) =(1+dlA) . Theonh~nxetbandftuthinh6mthu'dng<D(A)conc6b~cb~ng<p(dla).<p(a)-l.Ta dide'n <D(A)=@=Cl ... cr+s' M~nhd~du'<;1Cchungminh.0 12.15Djnh nghia : Nh6mcon .0 ::::;;<D(A)du'<;1cgQila ehinhtile trongnh6m<D(A)ne'u.0 c6 d~ng .0=.01 or+s, vdi.Qj ::::;;Cj, 1 ::::;;i ::::;;f + s. 12.16NhanxtH: Tit djnhnghlatrentatha'y.0chinhmctrong<D(A) khi vachikhi .0=.01...Or+s vdi.oj=.QnCi, 1 ::::;;i ::::;;f + s. * * 12.17Dinh If : Gift sa vanhR c6 h~ng6n djnh b~ng1va (1 +dA) ::::;;~::::;;(1 + A) * * vdichis6(il: (1+dA) ) Iebo~c«1+A) :~)Ie.Khid6,nh6mconil c~PvoiA khi vachikhinhomcon ill (1+dA)* chinhmctrongnhom<D(A) =(1 +A) */(1 +dA)* Changminh * D~t.0=il/(l +dA) ;.oi=.QnCj, 1 ::::;;i::::;;f + s. 65 1)Gii sa~ c~pvoi A, chungtachungminh.0=.01...Or+ s, Hi~n nhien, 01 ...Or+s~ n. Ngu'<;1c1~i,gii sa8lamQtphftntii'blltky trong.0,nghla1a8 E ~. Docos1!tu'dngungtrong(II) nencos1!phiintich * 0 =01...Or+ s ,vdi Oi E (1+ :i A) ,I ~ i ~r +s. Chungtachungminh 81E 11'\f1~ i ~r +s.Ta chungminhvoi tru'ongh<;1pi =1. - - * - B~t81=82 ... 8r+sthl 81 E (1+VIA) va8=81.81,Theoc6ngthuc7§3taco g~1,8-1)=Pf8-2-d =82Pf'812-d8f 8-2~2 -d) 1 2 '1 PI - d (a) 2 2 -2 2-2 P1.81 -d81 1 4P1 1 1-81 4PI d 1-81 (by= = + '-'P 1' + .. )2 2 2 4 2 4PI -d PI -d 81 PI -d PI Tachungminhl E (1 +dA)*. * Theof)inh1y8.4tac6 ~P1 E R.M~tkhac,do81 E ( 1+~A J ,81 E (1+P1A)*nen PI -d PI 2 d -2 2 d -2 ' 1- 81 E -.2A, 1- 81 E2PIA.HdnnU'a,1- 81 E -.4A, 1- 81 E 4pIAneu PI PI * V2(dA) ,;;1.Ta ill de-oY E (1 +d ~A) vahdooITaY E (1+dA)*oe-uV2(dA) ,;;1. Suy fa l E (1+dA)*.Theo(a)va(b)taco s:2- -1 ( S:-1)vI - Y g\P1,v Suyra 8i=y-2g~1.8-1j. Tudi~uki~ndachungminhtacoy-2E (1+dA)*. M~tkhac,do 8 E ~ c~pvoi A, ~ ~(1 + A)* ~no(A) nen theoBinh 1y12.4va 12.5taco g~l,8-1j E ~. Suyra 8i En. Tu'dngt1!, 81E ~voi2 ~i ~r +s. 66 * - Tiep IDeotagiii sa (~ : (1 +(dA) ) Ie, nghlaHicarda Ie. Vi 8 E Q nen t6n . ",.t' 0 h.. ~ ~4ll (~4 ~4 ) ll d " ki d h J ,00t.pmQtson ;?: saoc 0 v =v =VI .,. Vr+s . TIT leU ~n acling illl - tasuyradU<;1c8 E .01...Oc+s =>.0~.01...Oc+s Giii sa((1+A)* :~)Ie,tadac6 8{E~, 1~i ~r +s. * . Do((1+A) :~) Ie nen8iE ~,1-~i ~r +s.Suyra - - 8 =81... 8r+sE 01 ..,.or+s =>n ~.o1'" Or+s =>.0=.01...Or+s'V~y0 chinhde trongcI>(A) 2)Giasanh6m0 =~/(1+dA)* chinhtik trongcI>(A).Ta chungminh~c~p d6ivdiA. TITtich.0=01 ...Or+s tac6 ~=~1...~r+s * vdi ~i= An (!+;A) . * M~tkhac IDeoBinh 19'12.4va Binh 19'10.1thl nh6mcon(1 +dA) c~pd6i vdi A. Donh6m!huangCi=(I +: ArfI +dA)' Iii nh6ill xiclic nen IDeoBjnh Iji 12.9tac6 ~i c~pd6i A, 1 ~i ~r + s. Theo Binh 19'12.8ta c6 ~ c~pd6i vdi A. Binh 19'dU<;1cchung minh. 0 TITBad~12.12vaBinh19'12.17tasuyratn;(ctiepcaedinh19'sail: 12.18Binh Iy : Giii sah.~mgandinhcuavanhR b~ng1.Khi d6c6mQttu'dngung 1-1 gifi'acaenh6mcon~~Oo(A)c~pd6ivdiA c6chis6(Oo(A): ~)Ie vdicae nh6mconchiOOHieca'pIe cuanh6mcI>(A). 67 12.19Dinh If : Gia sah~ng6ndinhcuavanhR b~ng1,1:; la s6caetides6Ie cuaPi - 1, 1 ::;;i ::;;f. Khi do s6caenhomcon1:1::;;Qo(A)c~Pd6i vdi A co chi s6 (Qo(A): 1:1)Ie b~ng'Cl... 'Cr.2s. ~