抽象代数 孟道骥版 习题解答 第四章

Chapter44.1 Ӯ1. G 4. G 4 Klein K4 .đ Ϝ4 S4 . .(i)G 4 đ G 4 .(ii)G 4 đ ∀a∈G,a2=e.Ֆ ∀a,b∈G,(ab)2= e,, ab=(ab)−1=b−1a−1=ba, G Abel đ G∼=K4.2. G 6. G 6 S3 .G с 3 đ Ԣ 2 đ сAbel đ a=b∈G, a=e,b=e, a,b 4 đ .G с 2 đ Ԣ 3 đ |G| đ .G 2 a, 3 b.1):a,b đ ab 6 đՖ G= ab 6 .2):a,b҂ đ G 6 . G k 3 đj 2 đ2k+j+1=6, (k,j)=(2,1) (1,3). k=2, G3 {x,x−1,y,y−1}. xy҂ 3 đ xy2 đ yx đ xy=yx, x,y 9 đ. (k,j)=(1,3). թ G S6 ֆ ϕ,҂ ϕ(b)= (1,2,3), ϕ(a)=σ. G 3 đ σ(1,2,3)σ−1= (σ(1),σ(2),σ(3)), {σ(1),σ(2),σ(3)}={1,2,3}. σ (1,2,3)҂ đ ҂ σ(1)=1,σ(2)=2,σ(3)=3,҂ σ(1)=1,σ(2)=3,σ(3)=2.α=456σ(4)σ(5)σ(6)σ=(2,3)α, σ2=e, α2=e.σ,(1,2,3) ={(1,2,3),(1,3,2),e,(2,3)α,(1,2)α,(3,1)α} S36465G ∼=S 3.3. G r =st đH G t .H ={g s |g ∈G }={h ∈G |h=e }.G = g 0 , {g s |g ∈G }={g s 0,g 2s 0,···,g ts 0},{h ∈G |h t =e }={g s 0,g 2s 0,···,g ts 0}, {g s 0,g 2s 0,···,g ts 0} G t đ G t . G ={g s |g ∈G }={h ∈G |h t =e }4. G đa,b ∈G.ӫ[a,b ]=aba −1b −1 a,b .{aba −1b −1|a,b ∈G } Ӯ G (1)ӫ G. :1) α∈Aut G , α(G (1))=G (1);2) H G. G/H Abel ԉ H ⊇G (1). 1)α(G (1))=α( {aba −1b −1|a,b ∈G } )= {σ(a )σ(b )σ(a )−1σ(b )−1|a,b ∈G } =G (1).2)G/H Abel ⇔(G/H )(1)={e }⇔G (1)⊆H .5. S G Ӯ đ ϕ,ψ G H đ ϕ(x )=ψ(x ),∀x ∈S. ϕ=ψ. ∀a ∈G , G = S , a =y 1y 2···y n , y i ∈S y −1i ∈S . ϕ(x )=ψ(x ),∀x ∈S , ϕ(x −1)=ψ(x −1),∀x ∈S ,Ֆ ϕ(y i )=ψ(y i ),∀1≤i ≤n , ϕ(a )=ψ(a ), ϕ=ψ.6. H G đ H =G . G = G −H . H =G ∃a ∈G , GH , aH ∩H =∅, aH ⊇H , G −H ⊃H ∪(G −H )=G , G = G −H .7. G đ G с 2 . G k m čm >1Ďđ m kϕ(m ) đ ϕ . m đ ϕ(m ) đ Ԣ . |G | đ с đՖ թ 2 .8. α∈S 3 ҂ . α= 1234567836548271α= 1234567836548271=(1358)(26).669. S n= {(12),(23),···,(n−1n)} .. n=2 đ Ӯ . n=k Ӯ .n=k+1 đ{(12),···,(k−1k)} =S k(1k)(k k+1)(1k)=(1k+1){(12),···,(k−1k),(k k+1)} = {(12),···,(1k),(1k+1)} =S k+1{(12),···,(n−1n)} =S nn=k+1 Ӯ .Ֆ n∈N Ӯ .10. (i1i2···i r)−1=(i r i r−1···i1).(i1···i r)(i r i r−1···i1)=e11. ∀σ∈S n,σ(i1i2···i r)σ−1=(σ(i1)σ(i2)···σ(i r)).j=1,2,···,r−1,(σ(i1i2···i r)σ−1)(σ(i j))=σ(i1i2···i r)(i j)=σ(i j+1),j=r,(σ(i1i2···i r)σ−1)(σ(i r))=σ(i1i2···i r)(i r)=σ(i1),j=r+1,···,n,(σ(i1i2···i r)σ−1)(σ(i j))=σ(i1i2···i r)(i j)=σ(i j),∴σ(i1i2···i r)σ−1=(σ(i1)σ(i2)···σ(i r)).12. G 2k, k k>1. G с 2 .Cayley ğG L G|G|=2k թ 2 a∈G, ∀x∈G,L a(x)2=x,L a k đ k đՖ L a đ L G թ2 đ G .6713. σ r ⚷ . ğ1)σk Ԋ Ĥ2) σk đσk Ӊ Ĥ1) ӫ đσk ⇔σk Ӊ 1 r ⇔σk 1 r ⇔(k,r )=1 r .2) (k,r )=1 ,σk Ӊ r ; (k,r )=r đσk Ӊ 1.14. σ r ⚷ đ k |r. σk k Ӊ r/k ҂.n =r/k, n ∈N ., σ=(α1α2···αr )= α1α2···αr −1αrα2α3···αr α1σk = α1α2···αr −k αr −k +1···αr αk +1αk +2···αr α1···αkğσk =(α1αk +1α2k +1···α(n −1)k +1(=r −k +1))(α2αk +2α2k +2···α(n −1)k +2(=r −k +2)···(αk α2k ···αnk (=r )), ҂ đ Ӊ n , r/k ,∴σk k Ӊ r/k ҂ .4.21. A 1A 2A 3A 4 ш љ s 1,s 2,s 3,s 4. ш љ M 1,M 2,M 3,M 4. ш m 1,m2. d 1,d 2. O .č ĎG A 1A 2A 3A 4҂э ЌӉэ . ӈ GX ={A i ,M i ,s i ,m i ,d i ,O }. X .A 1A 2A 3A 4҂э ЌӉэ э č Ďđѩ A 1A 2A 3A 4҂эč ҂эđ ѓ эĎđ ӈ X ∀g ∈G,g R 2 ЌӉэ đg X P э э g (P ).∴X ğ{A 1,A 2,A 3,A 4}∪{M 1,M 2,M 3,M 4}∪{s 1,s 2,s 3,s 4}∪{m 1,m 2}∪{d 1,d 2}∪{O }.G 8 đ a =0−110č O π/2Ďđb ={ m 1 } Ӯđa 4đb 2.682. G X đY X đF Y ={g ∈G |g (Y )=Y }X Z đ ∃g ∈G Z =g (Y )đ ӫZ Y G .1)F Y G Ġ2)F g (Y )=ad g (F Y );3) G đ X Y [G :F Y ] . 1)∀g 1,g 2∈F Y , g 1g −12(Y )=g 1(Y )=Y , g 1g −12∈F Y , F Y G .2)∀g 1∈F Y đ gg 1g −1(g (Y ))=g (Y ), F g (Y )⊇ad g (F Y ). F Y =F g −1(g (Y ))⊇ad g −1(f G (Y )), F g (Y )⊆ad g (F Y ),Ֆ F g (Y )=ad g (F Y ).3) T ={g (Y )|g ∈G }. G T ğ∀Z ∈T,g (Z )={g (z )|z ∈Z }. Y F Y ,g T đ |G |<∞, X Y |T |=[G :F Y ].3. G H n . H Ї G N [G :N ]|n !.G G/H đ ԛG S n η,ker η= x ∈G/HF x , F eH =H , N =ker η. N ⊂H , NG ,[G :N ]|n !.4. p G đ H G [G :H ]=p . H G .p G đ G K,[G :K ] ҂ p , թ G N , N <H , [G :H ]|p !, [G :N ]=p , H =N , H G .5. G č ӫ ĎđH G đ[G :H ]=m . G ҂ m . H G .d =min {o (g )|o (g ) g đg ∈G , g =e }, d đ d ≥m . m =1, H G. m >1, g ∈G g ∈H , s =min {t |gt ∈H }, [G :H ]≥s , (s,o (g ))>1, s ≥d ! m ≥d ,Ֆ m đ 3 H G .6. G đ ğ1) G ={e } đG G ϴ .2) C (G )={e } đG G ϴ . 1) Oe ={e } ğG G ϴ ⇔G ={e }.2) .69 7. ԛS3,S4 .S3 3 đ љ (1),(12),(123) սіđC(1)= {(1)},C(12)={(12),(13),(23)},C(123)={(123),(132)}.8. S3 nn=n1+n2+···+n q,n1≥n2≥···n q≥1 .σ∈S n,σ(i1,···,i r)σ−1=(σ(i1),···,σ(i r))σ,η∈S n,σ η σ,η Ӯ҂ . S n n ğ σ∈S n,σ=σ1σ2···σk,σ1,σ2,···,σk k ҂ . |σi|і σi Ӊ đk|σi|=ni=1|σ1|≥|σ2|≥···≥|σk|. σ սі n ğn=|σ1|+|σ2|+···+|σk|..9. H G . ğ1)H G Ġ2)C h⊆H, h∈H;3)C q∩H=∅,g∈H..10. H G đ ğ1) ∀g∈G,H1=gHg−1 G čӫ H Ď.2) G N G(H)={g∈G|gHg−1=H} G HN G(H).čN G(H)ӫ H G đ ӫ H .Ď3)G H [G:N G(H)].4)H G⇐⇒N a(H)=G.1)∵ad g∈Int G,∴ HэӮG=ad gG gHg−1.2) X={gHg−1|g∈G}, G X ğg(K)=gKg−1,∀K∈X.70N G (H )=F H , H N G (H ) .3)2 đ G H|X |=[G :N G (H )].4)H G ⇔|X |=1⇔[G :N G (H )]=1⇔G =N G (H ).11. H G č H =G Ď. G = g ∈G gHg −1. 3 | g ∈G gHg −1|≤[G :N G (H )](|H |−1)+1≤[G :H ]|H |+1−[G :H ]=|G |+1−[G :H ]<|G |, G = g ∈GgHg −1.12. H G .GC G (H )={g ∈G |gh =hg,∀h ∈H }ӫ H G , ӫ H . :1)C G (H )=h ∈HC G (h );2)C G (H ) N G (H ). .13. H,K G .∀k ∈K ,ӫkHk −1 H K- . H ҂K - [K :K ∩N G (H )]. 10 2 K đ F H =K ∩N G (H ), H ҂ K − [K :K ∩N G (H )].14. θ G , θ(C (G ))=C (G ). .15. Aut G G ,C (G )={e }(e G ). C (Aut G )={id G }. θ∈C (Aut G ), ∀g ∈G,af gθad g −1=θ.Ֆ ∀h ∈G,ad gθad g −1(h )=θ(h ), (gθ(g −1))θ(h )(gθ(g −1))−1=θ(h ), θ(g )=g , θ=id G ,ՖC (Aut G )={id G }.16. a G . I ={g ∈G |α(g )=g −1}. :1) |I |>34|G |, G Abel ;2) |I |=34|G |, G Ї 2 Abel .71 17. G X , ∀x1,x2,y1,y2∈X,x1=x2,y1=y2,∃g∈G y1=g(x1),y2=g(x2). ӫG X .Π(X)={X1,X2,···,X k,···}X , g(π(X))={g(X1),g(X2),···,g(X k),···}, g(π(X))=π(X) ԉ π(X)={X} π(X)={{x}|x∈X}..ԉ .с . i |X i|>1,҂ x1,x2∈X i, ∀y=x1,x2, G X đ g, g(x1)=x1,g(x2)=y, g(π(X))=π(X), y∈x i,Ֆ X i=X, π(X)={X}.∀i,|X i|=1, π(X)={{x}|x∈X}.18. n≥2 ,S n {1,2,···,n} .S n .19.n ,A n {1,2,···,n} ?n=2,3 đA n {1,2,···,n} .n=2 3 đA n {1,2,···,n} ҂ .n=1 đ .n≥4 đ ∀x1,x2,y1,y2∈{1,2,···,n},x1=x2,y1=y2,թ σ∈S n y1=σ(x1),y2=σ(x2), x3,x4 x1,x2,x3,x4 ҂đ σ1=(σ(x3),σ(x4))σ, σ1,σ2 с A n, σ(x1)=y1,σ(x2)=y2, A n {1,2,···,n} .4.3Sylow1. p . p2 . ,p2 .C(G)={e}, p2 .1)G p2 đ G p2 .2)G p2 đ Ԣ đ p . a=e,b∈ a , G= a ⊗ b ∼=Z p⊕Z p. đp2 .2. p ,G=<a,b,c>.a,b,ca p=b p=c p=e,ab=ba,ac=ca,bc=cbe G . :G p3 , Ԣp.72 3. p ,F p l−1 , F G.:“ p G , H G , [G:H]=p, H G”.Ч p G ,F<G, [G:H]=|G|/|F|=p l/p l−1=p. ՎF G.4. ԛS4 Sylow3− .|S4|=24=3×23,∴S4 Sylow3− 3 , k . k≡1(mod3) k|8,Ֆ k 1 4.(123)∈S4, (123) 3 ,∴ (123) ={(1),(123),(132)} Sylow3− .(14)∈S4, Sylow (14) (123) (14)−1= {(1),(234),(243)} Sylow3− .,(24) (123) (24)−1={(1),(134),(143)},(34) (123) (34)−1= {(1),(124),(142)} Sylow3− .∴k=4.S4 Sylow3− :p1={(1),(123),(132)},p2={(1),(234),(243)},p3={(1),(134),(143)},p4={(1),(124),(142)} ( : ԛ4 3 (123),(124),(134),(234) .)5. 56 148 ҂ ֆ .G 56đ K G Sylow7− đ K≡1(mod7) K|8, K=1 8. K=1, G ֆĠ K=8, 7 8×(7−1)=48 đ G 8 đ G թ 8 Sylow2− đ 8 Ӯ Sylow2− đ 56 ֆ. |G|=148=22×37, Sylow37 đ 148 ֆ. 6. G 353. G 125 .Sylow G 125 Sylow5− đ 125 .7. G p l m,p , p>m(m=1). G҂ ֆ .Sylow p− .8. p,q ,p<q,p (q−1). pq .Sylow đpđq đՖ .9. 255 .255=3×5×17, Sylow ğSylow17− đSylow5− 1 51 đSylow3− 1 85 .73 Sylow17− P17= a , Sylow3− P3= b ,Sylow5− P5= c . P17 đ P5×P17 G đ P5×P17 đ ađc . P5×P17 8564 đ Sylow5− ҂ đ 51 đՖ 54×51204 đՖ 64Đ204<255, . Sylow5− .P3×P5,P3×P17 đՖ ađbđc đ G= abc 255 .10. H G ,p |G| , p [G:H]. HЇG Sylow p− .P [G:H] HЇ G Sylow p− đ P1, H G, gP g−1⊆H,∀g∈G,Ֆ HЇ G Sylow p− .11. G p l m,p ,(p,m)=1, m<2p. G Sylowp− p l−1 .G Sylow p− P, [G:P]=m, P GN [G:N]|m!, [G:P][P:N]|m!, [P:N] pՑ đ p||m!, [P:N]=1 p,Ֆ G Sylow p− p l−1 .12. p G , P G Sylow p− .N G(N G(P))=N G(P).13 ԛ.13. P G Sylow p− . G H⊇N G(P). N G(H)=H.H k Sylow p− đ P,P2,P3,···,P k. g∈N G(H), gHg−1=H, gP g−1⊆H, gP g−1=P j,h∈H hP h−1=P j,Ֆ (h−1g)P(h−1g)−1=P, h−1g∈N G(P)⊆H, g∈H,N G(H)=H.14. G Abel p− , [G:C(G)]≥p2.G Abel p− đ g∈G, g∈C G(G).g C(G) Abel đ [G: g C(G)]≥p, [ g C(G):C(G)]≥pՖ[G:C(G)]≥p2.15. P G Sylow p− đN G . N∩P,P N/N љ N G/N Sylow p− .H N Sylow p− đ g∈G gHg−1⊆P. N G, gHg−1⊆N,Ֆ gHg−1⊆N∩P. |H|= |N∩P|, |N∩P|=|H|,Ֆ N∩P N Sylow p− .74 p l1|||N|,p l|||G|, p l−l1|||G/N|. Ч P N/N∼= P/P∩N, p l−l1|||P N/N|, P N/N P G/N Sylow p− .ԛp l−l1|||P N/N|, p l−l1|||P/P∩N|, p l1||P∩N|, P∩N N Sylow p− .4.4 ֆ1. G e đ ğ(a)GG=G0⊃G1⊃G2⊃···⊃G n−1⊃G n={e}a.G i G i−1,1≤i≤n;b. G i−1/G i ֆ .(b) G đ G i−1/G i .1) n=pα11pα22···pαk k,p i đ l(n)=α1+···+αk.(G i/G k)/(G i+1/G k)∼=G i/G i+1, l(|G|) .2) G đ G i−1/G i Abel đ G i−1/G i ֆ đG i−1/G i .2. G Abel đ G 2 .n=|G|,n=2l0p l11···p l k k,2,p1,···,p k ҂ . թ G H, ğ2l0−1|||H|. P i G Sylow p i− đ1≤i≤k. G1=HP1···P k, [G:G1]=2.3. (123) A3 C(123).A3=<(123)>, Abel đ C(123)={(123)}.4. (123) A4 C(123).đ(123) S4 đ {(123),(132),(124),(142),(134),(143),(234),(243)}, Ađ C(123)⊆A.ՑđA α C(123)⇔∃σ∈A4, σ(123)σ−1=α, (σ(1)σ(2)σ(3))=α.đ(123)∈C(123).(124)=(σ(1)σ(2)σ(3))⇒σ=(34)∈A4,∴(124)∈C(123).(142)=(σ(1)σ(2)σ(3))⇒σ=(243)∈A4,∴(142)∈C(123).ԛ(143),(132),(234)∈C(123),(134),(243)∈C(123)∴C(123)={(123),(142),(234),(243)}755. n≥3,r,s {1,2,···,n} đA n=<{(r s t),t=r,s}>.đ n≥3 đA n 3− Ӯđ A n⊇< {(r s t),t=r,s}>.A n⊆<{(r s t),t=r,s}> .∀α∈A n,α і đ α=α1α2···αl.αi 2 ,i=1,2,···,l.ğ(1)(i j)(i j)=id=(r s t0)3,t0=r,s.∴(i j)(i j)∈<{(r s t),t=r,s}>.(2)(i j)(i k)=(i j)(i r)(i r)(i k)=(i r j)(i k r)=(r i)(r j)(r k)(r i)=(r i)(r s)(r j)(r k)(r s)(r s)(r i)=(r s i)(r j s)(r s k)(r i s)= (r s i)(r s j)2(r s k)(r s i)2, i,j,k=r,s,(i j)(i k)∈<{(r s t),t= r,s}>.(r j)(r k)=(r j)(r s)(r s)(r k)=(r s j)(r s k)2, j,k=s.(i r)(i k)=(i k r)=(r k)(r i)=(r k)(r s)(r s)(r i)=(r s k)(r s i)2,k=s.(i r)(i s)=(i s r)=(r s)(r i)(r s i)2, (s j)(s k),(i s)(i k)∈< {(r s t),t=r,s}>.đ ∀i,j,k,(i j)(i k)∈<{(r s t),t=r,s}>.(3)(i k)(j l)=(i k)(i j)(j i)(j l), ∀i,k,j,l,(i k)(j l)∈< {(r s t),t=r,s}>.đ αi∈<{(r s t),t=r,s}>,∴α∈<{(r s t),t=r,s}>.Ֆ A n⊆<{(r s t),t=r,s}>.6. G Abel ֆ đ |G|≥60.G n đ n<60, ҂ Abel . 2 3 5 7>60 n 3 ҂ .1Ďn đ n=p l. G҂ Abel đ l>1,C(G)= G, C(G) G, G ֆ.2)n 2 ҂ đ n=p a q b,p<q,p,q .A)p≥3, 60>n>3a+b, a+b≤3. a=1 đ Sylow q− đ ֆ. a=2 đn=32∗5,Sylow5− đG ֆ.B)p=2. q>7, p a<60/11<q, Sylow q− .G76ֆ. q ≥7, q =3,5,7. đ a =1 đSylow q − đG ֆ. a ≥2 đn ğ22×3,22×32,22×5,22×7,23×3,23×5,23×7. n =22×5 22×7 đSylow q − đ G ֆ. n =22×3 đ 3 ь G ֆ. n =22×32 23×3 đ Sylow 3− 4 1 . 4 đ X ={P 1,P 2,P 3,P 4} G Sylow 3− đ G X đ ԛG S 1×1 σ, G/Ker σ S 4 . n =22×32, |G/Ker σ|≤|S 4|=24, Ker σ={e }. G X đ Ker σ=G, Ker σ G . G ֆ. n =23×3, Ker σ={e }, G ֆĠ Ker σ={e },Ֆ G ∼=S 4, A 4 G G ֆ. n =23×5 đ Sylow 5− đ G ֆ. n =23×7 đ G ֆ. đ |G | 2 ҂ đG ֆ.3)n 3 ҂ . n/2×3×5≤1,n <3×5×7, n =2qr,2,q,r ҂ . 2||n , G с H [G :H ]=2,Ֆ G ֆ.ğ |G |<60, đ G ҂ ֆ . G Abel ֆ đ |G |≥60.7. G 60 ֆ đ G A 5 .H G đX ={gHg −1|g ∈G }, G X . G ֆ đ . n =[G :N G (H )]=|X |, G S n đ n ≥5,[G :H ]≥5. n =5, G S n đ G ∼=A 5. G Sylow 2− đ n 2 3đ5đ15 đ đn 2≥5: n 2=5, G ∼=A 5, n 2=15, 5 6×(5−1)=24 ğс 2 Sylow 2− ҂ {e }, 2 A. C G (A )Ї Sylow 2− đ 4|C G (A ),|C G (A )|>4, |G :C G (A )|≥5,Ֆ |C G (A )|=12,[G :C G (A )]=5. G ֆ đ N G (C G (A ))=C G (A ), [G :N G (C G (A ))]=5 G ∼=A 5. G ∼=A 5.8. G =SL(3,Z 2), Z 2 1 3 . G 168 ֆ .Z 2 1 đ SL(3,Z 2)=GL(3,Z 2). |GL (n,Z p )|=n −1 i =1(p n −p i ), |G |=(23−1)(23−2)(23−22)=168=23×3×7. Z 2 Ց ҂ն 3 ҂ ğx,x −1,x 2+x +1,x 3+x +1,x 3+x 2+1. Z 2 n77đSL(3,Z 2) 6 ѓ ğ⎛⎝100010001⎞⎠⎛⎝110010001⎞⎠⎛⎝110011001⎞⎠⎛⎝001101010⎞⎠⎛⎝001100011⎞⎠⎛⎝100001011⎞⎠ љ A 1,A 2,A 3,A 4,A 5,A 6, љ ğ(x −1),(x −1)2,(x −1)3,x 3+x +1,x 3+x 2+1,(x −1)(x 2+x +1). G A 1,A 2,···,A 6 սі 6 .A i A 1,A 2,···,A 6 љ ğ1đ2đ4đ7đ7đ3.235 8 {P (1×i,j )|1≤i,j ≤3,i =j } =G . 2 2 đ 2 3 . H G, H =G đ H 2 . đс 2 đ |H ||3×7. |C (A 2)|=8,|C (A 3)|=4,|C (A 6)|=3. 2 168/8Ģ21 ,4 42 đ3 56 đՖ 7 168−21−42−56−1=48 . Sylow p − ğ H 3 đ 56 3 Ġ H 7 đ 48 . |H |≤21, |H |=1. G ֆ .9. G G e đ C (G )={e }. :C (Aut G )={id }.2.15 .10. G Abel ֆ đA =Aut G. A . Aut A =Int A.4.51. p,q đ p <q . A p đ B q đG A B . ğ(a)G A B Ч Ġ(b) q p ҂ 1đ Վ đG pq Ġ(c) q ≡1(mod p ), թ A B G .Վ G Abel .1) N G, N ∼=B. G Sylow p − P đ P ∩N ={e }, P N =G. G A B Ч .2) q ≡1(mod p ), Sylow p − đ P , P G, P N =G. Վ đG pq .3)782. A đB љ m,n đ A B Ĥč đ ӫ .Ď3. A C (A )={1}, Aut A =Int A , ӫA . B A .C (A )={1}, C G (A )∩A ={e }, ∀g ∈G,ad g ∈Aut A ; Aut A =Int A, a ∈A, ad g =ad a, a −1g ∈C G (A ). G =C G (A )A. C G (A ) đ C G (A ) G , .4. N 1,N 2,···,N k G đ G N 1,N 2,···,N k ԉ ğ(a)G =N 1N 2···N k ;(b)N i G,1≤i ≤k ;(c)( j =iN j )∩N i ={1},1≤i ≤k . ԉ ğ 1đ2 Ӯ ∀g ∈G,g =n 1n 2···g k ,n i ∈N i і . m i ∈N i , g =n 1n 2···g k =m 1m 2···m k .(∗) m −11n 1=m 2m 3···m k n −1k n −1k −1···n −12=(m 2···m k )(n 2···n k )−1.∵N i G,1≤i ≤k,∴N 2N 3···N k G, m 2···m k ∈N 2N 3···N k ,n 2···n k ∈N 2N 3···N k , (n 2···n k )−1∈N 2···N k ,∴(m 2···m k )(n 2···n k )−1∈N 2···N k , m −11n 1∈N 1,∴m −11n 1∈N 1∩( j =1N j )={1}.Ֆ n 1=m 1. (∗) n 1,m 1, đ Ց n 2=m 2,n 3=m 3,···,n k =m k ∴g і đՖ ԉ .с ğ∵G N 1,N 2,···,N k đ∴1,2Ӯ . ∀g ∈G,g =n 1n 2···N k і . a ∈N 1∩( j =1N j ). n i ∈N i ,i =1,2,···,k a =n 1=n 2···n k , a =n 11·1·1····1 k −1=1·n 2n 3···n k , і n 1=n 2=···=n k =1,∴a =1 N 1∩( j =1N j )={1}. N i ∩( j =iN j )={1},1≤i ≤k. 3Ӯ .5. G =N 1N 2···N k . (j =i N j )⊆C G (N i ).79N i ∩N j ={e },N i N i N j ,N j N i N j , N i N j đՖ ( j =i N j )⊆C G (N i ).6. G =A ⊗B, N B . ğ(a)N G Ġ(b)G/N A ×(B/N ) .(a)∀g ∈G,n ∈N , G =A ⊗B ∃a ∈A,b ∈B . g =ab .∴gng −1=abnb −1a −1,∵N B,∴bnb −1∈N ⊆B. G =A ⊗B a (bnb −1)=(bnb −1)a,∴gng −1=bnb −1aa −1=bnb −1aa −1=bnb −1∈N,∴N G(b) f :A ×(B/N )−→G/N,f ((a,bN ))=abN. đ (a,b 1N )=(a,b 2N ), b −11a −1ab 2∈N , (ab 1)−1(ab 2)∈N,∴ab 1N =ab 2N,∴f . Ցđ f ((a 1,b 1N ))=f ((a 2,b 2N )), a 1b 1N =a 2b 2N,(a 1b 1)−1a 2b 2∈N,∃n 1∈N, b −11a −11a 2b 2=n 1.∵G =A ⊗B,∴b −11(a −11a 2)=(a −11a 2)b −11, (a −11a 2)(b −11b 2)=1n 1. 1∈A,n 1∈N ⊆B . G с a −11a 2=1,b −11b 2=n 1, a 1=a 2,b −11b 2∈N ∴b 1N =b 2N , (a 1,b 1N )=(a 2,b 2N ),∴f ֆ .Ցđ∀gN ∈G/N,∃a ∈A,b ∈B , g =ab , gN =abN (a,bN ),∴f .đf ((a 1,b 1N )(a 2,b 2N ))=f ((a 1a 2,b 1b 2N ))=a 1a 2b 1b 2NA B a 1a 2b 1b 2N =a 1b 1a 2b 2N =a 1b 1N ·a 2b 2N f ((a 1,b 1N )(a 2,b 2N ))=f ((a 1,b 1N ))f ((a 2,b 2N )),∴f . f A ×B/N G/N đ G/N A ×B/N .7. A,B G đ G =AB. G/(A ∩B )=A/(A ∩B )⊗B/(A ∩B ).A,B G đ A/(A ∩B ),B/(A ∩B ) G/(A ∩B ) đ (A/(A ∩B ))∩(B/(A ∩B ))=(A ∩B )/(A ∩B )={id },A/(A ∩B )×B/(A ∩B )=G/(A ∩B ), G/(A ∩B )=A/(A ∩B )⊗B/(A ∩B ).80 8. A,B G , A∩B={1}. G (G/A)×(G/B).fG−→G/A×G/Bf(g)=(gA,gB), f .(g1A,g1B)=(g2A,g2B), g1A=g2A,g1B=g2B,∴g−11g2∈A∩B,∴g−11g2=1,∴g1=g2.Ֆ f ֆ đ f(g1g2)=(g1g2A,g1g2B)=(g1A,g1B)(g2A,g2B)= f(g1)f(g2).∴f ֆ . f:G−→f(G) .∴G f(G).f(G) G/A×G/B đ Ӯ .9. A,B G , G=AB;ab=ba,∀a∈A,b∈B. թA×B G .σ:σ(a,b)=ab .10. Y X đ G,A,B љ P(X),P(Y),P(X−Y) ӫҵo ” Ӯ čҕ 1.2 7Ď. G A×B .f:G−→A×B,f(M)=(M∩Y,M−Y). f .f(M)=f(N), (M∩Y,M−Y)=(N∩Y,N−Y), M∩Y= N∩Y,M−Y=N−Y.Ֆ M=(M∩Y)∪(M−Y)=(N∩Y)∪(N−Y)=N,∴f ֆ .∀A1∈A,B1∈B, M=A1∪B1, M∈G,f(M)=(M∩Y,M−Y)=(A1,B1).∀(A1,b1)∈A×B, A1∩B1. f .∀M,N∈G,f(M N)=((M N)∩Y,(M N)−Y)=((M∩Y) (N∩Y),(M−Y) (N−y)=(M∩Y,M−Y) (N∩Y,N−Y)=f(M) f(N).f .∴f .∴G A×B.11. G đ|G|=p a11p a22···p a k k,p1,p2,···,p k ҂. Sylow p i− P i G. G=P1⊗P2⊗···⊗P k.ğ G=(si=1N i)⊗(ti=1M i), G=N1⊗···⊗N s⊗···⊗M t.81 4.61. ğn≤4 đS n Ġn≥5 đS n .Ӯ 60 ֆ Abel đ60 . n≤4 đ|S n|<60, S n .n≥5 đ A n҂ đ S n҂ .2. GL(2,Z2),GL(2,Z3) .ğ |GL(2,Z2)|=|(22−1)(22−2)|< 60,|GL(2,Z3)|=(32−1)(32−3)<60, GL(2,Z2),GL(2,Z3) .3. G pqr, p,q,r ҂ . G .pq đ G Sylow q− Sylow r− , G Ġ G ҂թ Sylow q− Sylow p− đ Sylow r− pq đSylow q− r đ r q pq(r−1)+r(q−1)≥pqr−pq+rp>pqr=|G|, . pqr .4.(a) H,K G đ HK G.(b) R Gč҂ Ď ն đH G. H⊆R, G/R .(a) đH G,K G⇒HK G. 49 1.7.3 HK/KH/H∩K.∵H đ∴ H/H∩K .Ֆ HK/K.K đ HK đ HK G .(b) R H G đ 1 HK Gđ H R, R HR,Վ R G նđ∴H⊆R.π:G−→G/R đ∴kerπ=R. πG Ї R G/R. Ϝ đK⊇R,K G, K đ K π|K(K). π|K Kπ|K(K) .∴π|K(K) . đ∵π đ∴ G/RӮ π|K(K). K⊇R, K G. π|K(K)=K/R82 . R đ K . G Ї R G/K. K G/Kđ K=R. K1 G Ї R đс K1 R, ∵G҂ đ∴K1 G.Վ R p Gն o .∴G/R .5. G . ğ(a)G Ġ(b)∀H G,∃H1 H H/H1 Abel K={1};(c)∀H G,G/H Abel K={1}.1)⇒2)⇒3) .3)⇒1):G ն Hđ G/H .3Ď G=H, G .6. H G ն . N G(H)=H.N G(H)=H, g∈N G(H), g∈H. H N G(H), g H . g H/H∼= g /( g ∩H), g H/H đՖg H đ g H⊃H, H ն . N G(H)=H.7. n đ n ԉ n=2k,k∈N.n ҂ 2 p, թ Abel H:|H|=2p.H đ H҂ . n=2pk,k Kđ G=H×K, H҂ đ G҂ đ n đ n=2k. n=2k đ G .8. G .H G ն . N G(H)=H.G đ k∈N, T k=H, H⊃T k+1. [T k,H]⊆[T k,G]=T k+1, T k⊂N G(H),H=N G(H).č G Ϯ Ď.9. G .M G ն . M G. G Sylow.∵M G ն đ∴M G đ G đ M N G(M) . N G(M)=Gđ M N G(M)83 G,Վ M G ն .∴N G(M)=G,Ֆ M G.G Sylow P, 163 12 N G(N G(P))=N G(P).(∗)N G(P)=G, N G(P) G đ G đ∴N G(P) N G(N G(P)) đ (∗) .∴N G(P)=G.Ֆ P G, G Sylow .10. G ∀H G,C(G/H)={1}.G đ Ӊ đ H=G HG đC(G/H)={1}.∀H G,H=G, C(G/H)={1}. |G|<+∞, G Ӊ đ G .4.7Jordan-H¨o lder1. ԛ ZZ⊃20Z⊃60Z⊃{0},Z⊃49Z⊃245Z⊃{0}.Z⊃20Z⊃60Z⊃{0} Z⊃49Z⊃245Z⊃{0} љ ğZ⊃4Z⊃20Z⊃60Z⊃2940Z⊃{0},Z⊃49Z⊃245Z⊃980Z⊃2940Z⊃{0}. .2. Z60 Ӯ đѩ ..3. G Ց č ĎG=G1⊃G2⊃···⊃G r={1}.G i/G i+1 S i,i=1,2,···,r−1. G S1S2···S r−1..4. Abel G Ӯ G .Ӯ Abel .845. G Ӯ G .Gթ Ց G=G1⊃G2...G r=1,G i/G i+1 Abel đ1≤i≤r−1.G Ӯ G .6. G=G1⊃G2⊃···⊃G r={1} G Ӯ đ1đN G ֆ .G1⊇G2N⊇···⊇G r−1N⊇G r N⊇G r={1}҂ G Ӯ .G i/G i+1 G i N/G i+1N ϕ:ϕ(gG i+1)=gG i+1N,∀g∈G i. G i/G i+1 ֆ đ G i N/G i+1N={1} ֆ . G1⊇G2N⊇...G r−1N⊇G r N={1}. ҂ G Ӯ .7. ϕ G H .G=G1⊃G2⊃···⊃G r={1}G Ӯ .H=H1⊇ϕ(G2)⊇···⊇ϕ(G r)={1}҂ H Ӯ .G H ϕ ԛG i/G i+1 ϕ(G i)/ϕ(G i+1) ϕ:ϕ(gG i+)=ϕ(g)ϕ(G i+1),∀g∈G i. đ G i/G i+1 ֆ đ ϕ(G i)/ϕ(G i+1){1} ֆ . H=H1⊇ϕ(G2)⊇...⊇ϕ(G r)= {1} ҂ H Ӯ .8.R- ML1⊆L2⊆···⊆L n⊆···n L n+i=L n,i=1,2,···. ӫM đ ӫM Noether .R- ML1⊇L2⊇···⊇L n···n L n+i=L n,i=1,2,···. ӫM đ ӫM Artin .R- M Ӯ M Noether Artin .85R − M Ӯ đ M Ӯ đ M с Noether Artin . R − Noether Artin đ M 1=M . M đ M 1 M 2 M 1⊇M 2đ M 1/M 2 ֆ đ Ց đ M đ թ R ğM =M 1⊇M 2⊇...⊇M r ={0}, M i /M i +1,i =1,2,...,r −1 ֆ đ M Ӯ .4.8 ϶1. S,T G đ gSg −1⊆S,∀g ∈G. ğ(a)<S > G ;(b)< g ∈G gT g −1> G Ї T .1).2) g ( g ∈G (gT g −1))g −1= g ∈G (gT g −1),∀g ∈G ,g ∈G(gT g −1) G đ Ї T .2. X ={x 1,x 2,x 3}, {x 21,x 22,x 23} Ӯ F (X ) K F (X ) . F (X )/K G 1={x 1k 1x 2k 2x 3k 3|0≤k 1,k 2,k 3≤1} đ x 1,x 2,x 3∈G 1, F (X )/K =G 1, [F (X ):K ]≤8. F (X ) Z 2⊕Z 2⊕Z 2 ϕğϕ(x 1)=(1,0,0),ϕ(x 2)=(0,1,0),ϕ(x 3)=(0,0,1).ker ϕ⊇{x 12,x 22,x 32}, ker ϕ⊇K,đ [F (X ):K ]≥8đՖ [F (X ):K ]=8.3. S 4= (12),(13),(14) ,X ={x 1,x 2,x 3}. S 4 Ӯ x 2i ,(x i x j )3,(x i x j x k )2. i,j,k ҂ .4. S 4= (12),(13),(14) ,X ={x 1,x 2,x 3}, S 4 Ӯ x 2i ,(x i x i +1)3,(x i x j )2., j >i +1.5. 3 S n ,ѩ .6. 4 S n ,ѩ .。