刘绍学《近世代数基础》习题解答[1].pdf

1 1.1. P4,Ex1 OdetO = 1 O = ? cossin sincos ? O11 cos ?= 01 (1 cos)x siny = 0 l : (1 cos)x siny = 0 ? x y ? R2 ? x y ? , O ? x y ? = ? cosx + siny sinx cosy ? l ? x y ? l |(1cos)xsiny| 22cos O ? x y ? l |(1cos)(cosx+siny)sin(sinxcosy)| 22cos = |(1cos)xsiny| 22cos 1 ? x y ? O ? x y ? l sinxcosyy cosx+sinyx 1cos sin = 1 ? x y ? O ? x y ? ll 1 cos = 01 sinx (1 + cos)y = 0 l1: sinx (1 + cos)y = 0 ? x y ? O ? x y ? l1 l1? 1.2. P4,Ex2 , T(M)( ) = ( ) m M ( ) (m) = ( )(m) ( ) (m) = ( )(m) = (m) ( )(m) = ( )(m) = (m) ( ) (m) = ( )(m)( ) = ( )? 1.3. P4,Ex3 S(K)60; 120; 180; 240; 300 12? 2 2.1. P8,Ex1 F0, 1C ai+ bi2 F, ai, bi Q, i = 1,2 2 (a1+ b12) (a2+ b22) = (a1 a2) + (b1 b2)2 F (a1+ b12)(a2+ b22) = (a1a2+ 2b1b2) + (a1b2+ b1a2)2 F 1 a1+b12 = a1 a2 12b21 + ( b1 a2 12b21 2) F F0 ?= a = a1+ b12 F a1= a1 a2 12b21 + ( b1 a2 12b21 2) F F? 2.2. P8,Ex2 FQ F Aut(F : Q) Aut(F) Aut(F) Aut(F : Q)F Q Aut(F)(1) = 1n, (n) = n, (n) = n, (n1) = n1q = m n Q, m, n Z Z (m n) = (m n 1) = (m) (n1) = m n1 = m n Aut(F : Q)? 2.3. P8,Ex3 (1)x, y Qx+y2 = 0 x = y = 0 x, y Q0z Qzx, zy (zx, zy) = 1x, y0(x, y) = 1 x + y2 = 0 x2= 2y2 xx = 2k k2k2= y2y (x, y) = 1x = y = 0 (2)x, y Qx+y6 = 0 x = y = 0 x, yx + y6 = 0 x = y = 0 x, y0(x, y) = 1 x + y6 = 0 x2= 6y2 xx = 2kk2k2= 3y2y (x, y) = 1x = y = 0 3 (3)x, y Qx+y3 = 0 x = y = 0 x, yx + y3 = 0 x = y = 0 x, y0(x, y) = 1 x + y3 = 0 x2= 3y2 x3x = 3kk3k2= y2y3 (x, y) = 1x = y = 0 (4)x, y, z Qx+y2+z3 = 0 x = y = z = 0 x + y2 + z3 = 0 x2= (y2 + z3)2= 2y2+ 2yz6 + 3z2 2y2+ 3z2 x2+ 2yz6 = 0 (2)yz = 0y = 0z = 0 y = 0 x + y2 + z3 = 0 x + z3 = 0(3) x = z = 0 z = 0 x + y2 + z3 = 0 x + y2 = 0(1) x = y = 0 x + y2 + z3 = 0 x = y = z = 0 (5)a, b, c, d Qa+b2+c3+d6 = 0 a = b = c = d = 0 a + b2 + c3 + d6 = 0 (b2 + c3 + d6)2= (a)2 2b2+ 3c2+ 6d2+ 4bd3 + 6dc2 = a2 (4)bd = 0dc = 0 d = 0a + b2 + c3 + d6 = 0 a + b2 + c3 = 0(4) a = b = c = 0 b = 0c = 0a+b2+c3+d6 = 0 a+d6 = 0(2) a = d = 0 a + b2 + c3 + d6 = 0a = b = c = d = 0? 4 2.4. P8,Ex4 (1) 0, 1 Q(i) ak+ bki Q(i), k = 1,2 (a1+ b1i) (a2+ b2i) = (a1 a2) + (b1 b2)i Q(i) (a1+ b1i) (a2+ b2i) = (a1a2 b1b2) + (a1b2+ b1a2)i 0 ?= a1+ b1i Q(i) (a1+ b1i)1= a1 a2 1+b21 + ( b1 a2 1+b21 )i Q(i) Q(i) 0, 1 Q(i, 5) ak+ bki + ck5 + dk5i Q(i), k = 1,2 (a1+ b1i + c15 + d15i) (a2+ b2i + c25 + d25i) = (a1 a2) + (b1 b2)i + (c1 c2)5 + (c1 c2)5i Q(i, 5) (a1+ b1i + c15 + d15i) (a2+ b2i + c25 + d25i) = (a1a2 b1b2+ 5c1c2 5d1d2) + (a1b2+ b1a2+ 5c1d2+ 5d1c2)i +(a1c2+c1a2d1b2b1d2)5+(a1d2+d1a2+c1b2+b1c2)5i Q(i, 5) 0 ?= a + bi + c5 + d5i Q(i, 5) (a + bi + c5 + d5i)1 = a(a2+5c2+b2+5d2)5c(2ac+2bd) (a2+5c2+b2+5d2)25(2ac+2bd)2 + 5d(2ac+2bd)b(a2+5c2+b2+5d2) (a2+5c2+b2+5d2)25(2ac+2bd)2i + c(a2+5c2+b2+5d2)a(2ac+2bd) (a2+5c2+b2+5d2)25(2ac+2bd)2 5 + d(a2+5c2+b2+5d2)+b(2ac+2bd) (a2+5c2+b2+5d2)25(2ac+2bd)2 5i Q(i,5) Q(i, 5) (2)Ex2Aut(F : Q) = Aut(F) Aut(F), a + bi Q(i)(a + bi) = a + b(i) (i) i i = 1(i i) = (1) = 1(i) (i) = 1 (i) = i(i) = i (i) = i : Q(i) Q(i) a + bi ? a + bi 5 Q(i) (i) = i 1: Q(i) Q(i) a + bi ? a bi Q(i) Aut(F)Aut(Q(i) = I, 1IQ(i) Ex2Aut(E : Q) = Aut(E) Aut(E), a+bi+c5+d5i Q(i, 5) (a+bi) = a + b(i) + c(5) + d(5)(i) (i)(5) i i = 1(i i) = (1) = 1(i) (i) = 1 (i) = i(i) = i 55 = 5 (5 5) = (5) = 5 (5)(5) = 5 (5) = 5 (5) = 5 Aut(Q(i, 5) 4 I : Q(i, 5) Q(i,5) a + bi + c5 + d5i ? a + bi + c5 + d5i 1: Q(i, 5) Q(i,5) a + bi + c5 + d5i ? a bi + c5 d5i 2: Q(i, 5) Q(i,5) a + bi + c5 + d5i ? a + bi c5 d5i 3: Q(i, 5) Q(i,5) a + bi + c5 + d5i ? a bi c5 + d5i I,1,2,3 Aut(Q(i, 5) Aut(Q(i, 5) = I, 1,2,3 Aut(E : F)a Q,(a) = a, (i) = iAut(E : F) Aut(E)Aut(E : F)I,2 Aut(E : F) = I,2? 6 3 3.1. P11,Ex1 Sf= ? 123 123 ? , ? 123 321 ? ? 3.2. P11,Ex2 x3 1x2 f(x1,x2,x3) = x3 1x2+ x 3 2x1+ x 3 1x3+ x 3 3x1+ x 3 2x3+ x 3 3x2 ? 3.3. P11,Ex3 f(x,y) = 0K(a, b)f(a, b) = 0f(x,y)f(x,y) = f(y,x)f(b, a) = 0 (a, b)Kx y = 0(b, a)K Kx y = 0? 3.4. P11,Ex4 E2f = x2 2E Ef = x2 2 F2E F FQ F 2 F F a, b Q F, 2 F a + b2 F E FE2Ef = x2 2 ? 7 1 1.1. P17,Ex1 (G, )a Gx G ax = xa = e e(G, ) G ab = ac x(ab) = x(ac) (xa)b = (xa)c eb = ec b = c ba = ca (ba)x = (ca)x b(ax) = c(ax) be = ce b = c (G, )? 1.2. P17,Ex2 (1)(S,)x SxS = S e1 Sxe1= xy GSy = Sz G zx = y ye1= (zx)e1= z(xe1) = zx = y x SSx = Se2 Se2x = x y GyS = Sz Gxz = y e2y = e2(xz) = (e2x)z = zx = y e1= e2e1= e2= e(S,) 1 y SyS = Sy = Sy1,y2 S xy1= y2x = e y1= ey1= (y2x)y1= y2(xy1) = y2e = y2 y1= y2y (S,) (2)(S,)a S fa: S S, x ? ax faSSfa(S) = aSx,y Sfa(x) = fa(y)ax = ayx = yfaSSS faaS = S ga: S S, x ? xa gaSSga(S) = Sax,y Sga(x) = ga(y)xa = yax = ygaSSS gaSa = S (1) (S,) (A)? 1.3. P17,Ex3 1HGx,y H a,b G (a) = x, (b) = y, (a b) = (a) (b) = x y 1(x) = a, 1(y) = b, 1(x y) = a b = 1(x) 1(y) 1(H,)(G,)? 1.4. P17,Ex4 (1)(Z,) (i) 2 (ii) (iii)1a Z a 1 = a + 1 1 = a = 1 a (iv)a Za + 2 Za (a + 2) = a + (a + 2) 1 = 1 = (a + 2) a (v) (Z,) (2)ZZa,b Z(a + b) = a + b + 1 = (a + 1) + (b + 1) 1 = (a + 1) (b + 1) = (a) (b) (Z,+)(Z,)? 2 2.1. P22,Ex1 mnl 0 r n m = ln + r am= aln+r= anlar= (an)larar= e nr = 0n|m? 2.2. P22,Ex2 ab ba ab,baabn (ab)n= e eG (ba)n+1= b(ab)na = bea = ba (ba)n= e babam(ba)n= eEx1 m|n (ba)m= e (ab)m+1= a(ba)mb = aeb = ab (ab)m= e 3 abnEx1n|m m = n abba? 2.3. P22,Ex3 (1)H, KG H1= H, K1= K, HH = H, KK = K, (HK)1= K1H1= KH HKG HK = (HK)1= K1H1= KH HK = KH (HK)(HK) = H(KH)K = H(HK)K = (HH)(KK) = HK HKG (HK)1= K1H1= KH = HK HKHK HKG (2)HGa GaH = Ha GKHK = KH(1)HKG ? 2.4. P22,Ex4 S3= I, ? 123 132 ? , ? 123 213 ? , ? 123 231 ? , ? 123 312 ? , ? 123 321 ? G1= I G2= I, ? 123 132 ? G3= I, ? 123 321 ? G4= I, ? 123 213 ? G5= I, ? 123 312 ? , ? 123 231 ? S3 G2,G3,G4,G5G G5S3? 4 2.5. P22,Ex5 GInn(G)Aut(G) Ta Inn(G) Aut(G) Ta1 Inn(G) x G (Ta1)x = (Ta(1x) = (a(1x)a1) = (a)(1x)(a1) = (a)x(a)1= Ta Inn(G) Inn(G)Aut(G)? 3 3.1. P27,Ex1 :1= (itit1i2i1) t mm? 3.2. P27,Ex2 G =HG H = e = H ?= ee ?= b GG =l Z b = alHal= b1 HM = k|ak H, k N+ M ?= m = minMH = h Hk Zh = ak q,r Z,0 r 0 k = qm + r ak= aqm+r= aqmar= (am)qar ar= ak(am)q Hak H,am H,(am)q Har Hm r = 0h = ak= (am)qH =? 3.3. P27,Ex3 5 1) Gma Gam= eaG mnan= em | n (as) n (s, n) = (an) s (s, n) = e s (s, n) = ek Z(as)k= e = askann | (sk) n (s,n) | s (s,n)k ( n (s,n), s (s,n) = 1 n (s,n) | kas n (s,n) 2)1)a(s,n) n (s,n),n) = n (s,n) asa(s,n) 3) n (s,n) l,k Z (s,n) = ls + kn a(s,n)= als+kn= (as)l(an)k= (as)l =? 3.4. P27,Ex4 GG = e,a,b,b2,ab,ba |eabb2abba | e|eabb2abba a|aeabbabb2 b|bbab2eaab b2|b2abebbaa ab|abb2baaeb ba|babaabb2e ? 3.5. P27,Ex5 1)t (i j),i jj i j i = 1 6 j i = mj i = m + 1(i j) = (i i + 1)(i + 1 j)(i i + 1)(i + 1 j) (i j) 2)(i i+1) = (1 i)(1 i+1)(1 i)(1 2),(1 3),.,(1 n) Sn 3)3An3 (1 2 i)3 (i j k) = (1 2 k)(1 2 j)(1 2 i)(1 2 k)(1 2 j),i,j,k ?= 1,2 (1 i j) = (1 2 i)(1 2 j)(1 2 j)(1 2 i)(1 2 i),i,j ?= 1,2 (2 1 i) = (1 2 i)(1 2 i),i ?= 1,2 (2 i j) = (1 2 i)(1 2 j)(1 2 j),i,j,k ?= 1,2 (1 2 3),(1 2 4),.,(1 2 n)An? 4 4.1. P32,Ex1 b = a n 2 b ?= e,b2= eG2 f Aut(G)(f(b)2= f(b2) = f(e) = e f(b) = a n 2= b b Aut(G)? 4.2. P32,Ex2 B4 = Te,Ta,Tb,Tc Te= (e), Ta= (e a)(b c), Tb= (e b)(a c), Tc= (e c)(a b)? 5 5.1. P37,Ex1 = a|a GG a,b G ab ab 7 ab = abe abe abe = ab ab ab = ab ? 5.2. P37,Ex2 1)GHG x Gx xxx1= e H HGH ?= x, y Hxe1= x H,ye1= y Hx e y x y xy1 H y Hy1= ey1 HHH x, y Hy1 Hxy = x(y1)1 HHG HG HGG x Gxx1= e Hx x x, y Gx yxy1 Hyx1= (xy1)1 H y x x, y, z Gx y, y zxy1, yz1 Hxz1= (xy1)(yz1) Hx z G 2)GHG GGHG a G, x Ha a, x eax ae, ax a axa1 HaHa1 HHG HGG 8 HGG a, b, c, d G a b, c d ab1,cd1 HHG a(cd1)a1 H a(cd1)a1(ab1) = a(cd1)b1= ac(d1b1) = ac(bd)1 H ac bdG? 5.3. P37,Ex3 1) x Rx x R = ? xRx x, y Rx y ?= z x yz = n1a + x = n2a + yr x r = na + x = (n n1)a + n1a + x = (n n1)a + n2a + y = (n n1+ n2)a + y y x yr y r = na + y = (n n2)a + n2a + y = (n n2)a + n1a + x = (n n2+ n1)a + x x y x x = yR R x,y x + y = n1a + x + n1a + y = x + y R 2)C = ei|0 2x R (x) = ei 2 a x x = y(x) = (y) x = y x y = na (x) = ei 2 a x = ei 2 a (na+y) = ei2nei 2 a y = ei 2 a y = (y) (x) = (y)ei 2 a x = ei 2 a y ei 2 a (xy) = 1 2(xy) a = 2n x y = na x = y 9 ei Cx = a 2 R a 2 ( a 2) = e i2 a a 2= ei x, y (x) = ei 2 a x,(y) = ei2 a y (x + y) = ei 2 a (x+y) = ei 2 a xei2 a y = (x)(y) (, +)(C, )? 6 6.1. P42,Ex1 1)a, b (S)x, y S(x) = a, (y) = b, xy S ab = (x)(y) = (xy) (S) (S) a (S)x S(x) = aS x1 S, (x1) (S) a1= (x)1= (x1) (S) (S) (S) H(S)H 2)x, y 1(T)(x),(y) TTH (x1) = (x)1 T, x1 1(T) 1(T)1(T) (xy) = (x)(y) T, xy 1(T) 1(T)G 1(T)G THh H,t T hth1 ThTh1 T 10 a G,b 1(T)(a) H, (b) T, (a1) H (aba1) = (a)(b)(a1) T aba1 1(T) 1(T)G 3)a S Kerx S, y Ker, a = xy (a) = (xy) = (x)(y) = (x)e = (x) (S) a 1(S) S Ker 1(S) b 1(S)(b) (S)s S (b) = (s) (s1b) = (s1)(b) = (s)1(b) = (b)1(b) = e s1b Ker b = s(s1b) S Ker 1(S) S Ker 1(S) = S Ker? 6.2. P42,Ex2 L(G, H)L(G) SG(S)G S1, S2 L(G, H)(S1) = (S2),(S1) = (S2) S1= S2 x S1(x) (S1) = (S2)y S2 (y) = (x)(xy1) = exy1= s Ker = H S2 x = ys S2S1 S2 S2 S1 S L(G)1(S) L(G, H) (1(S) = (1(S) = S 11 1)S,T L(G, H) S T( H = Ker)x (T) y T S(y) = x (S)(T) (S) (T) (S)x T(x) (T) (S) y S(y) = (x),(xy1) = e, xy1 Ker = H S h H,xy1= h, x = hy ST S 2)SGSG(S)G x (S), y Ga S, b G(a) = x, (b) = y, (b1) = y1Sbab1 S yxy1= (b)(a)(b1) = (bab1) (S) (S)G (S)Gx (S), y G yxy1 (S) a S,b G(a) (S), (b), (b1) G, (bab1) = (b)(a)(b1) = (b)(a)(b)1 (S)s S (bab1) = (s), phi(bab1s1) = e, bab1s1 Ker H S h Hbab1s1= h, bab1= hs SSG 3)SG(S)GG/(S) : G G/(S) x ? (x) = (x)(S) G G/(S) Ker = S x S, (x) (S), (x) = (x) = (x)(S) = (S) x Ker S Ker x Kerx = (x)(S) = (S)(x) (S) s S(x) = (s), (xs1) = e G, xs1 Ker = H S h Hxs1= h, x = hs SKer S G/S = G/(S) ? 12 6.3. P42,Ex3 1)SG(S)H s S?(s) = (s) (S)? ?: S (S) ? 2)x S Ker?(x) = (x) = e (S),x Ker? S Ker Ker? x Ker?x S(x) = ?(x) = ex Ker x S Ker Ker? S Ker Ker?= S Ker 3)S/(S Ker)= (S) ?S(S)Ker?= S Ker ? 6.4. P42,Ex4 1)HGSG SH = HSP22,Ex3,(2)SHG x H y SH GHGyxy1 H HSH SHSx SH, y Sx H,yxy1 H, yxy1 S, yxy1 S H S HS 2)HSHSH/H : S SH/H x ? xH SH/H SSH/Hx,y S (xy) = xyH = (xH)(yH) = (x)(y)SSH/H xH SH/Hx SHs S, h Hx = sh xH = (sh)H = s(hH) = sHs S(s) = (x) = xH 13 SSH/H x Kerx S(x) = xH = Hx Hx S HKer S H x S Hx H, (x) = xH = Hx Ker S H Ker S H = KerS/S H = SH/H? 7 7.1. P46,Ex1 a G a ?= eam1m | pp m = pp= P? 7.2. P46,Ex2 HG2GH H, aHaH aH = H = Ha a Hx G xH = H x H Hx = H xH = Hx xH = aH xH Hx ?= H Hx = Ha xH = Hx HG? 7.3. P46,Ex3 1)a1 = a2, b1 = b2p | (a1 a2), p | (b1 b2) a1b1 a2b2= a1b1 a2b1+ a2b1 a2b2= b1(a1 a2) + a2(b1 b2) p | (a1b1 a2b2) a1b1 = a2b2 a, b, c Zp 0 (ab)c = abc = (ab)c = a(bc) = abc = a(bc) 1 Zp 0a Zp 0 14 a1 = 1a = a Zp 0 b Zp0b ?= kp(b, p) = 1l, m Z lb + mp = 1 lb + mp = 1, lb + mp = 1, lb = 1 lb Zp0p1p1 (Zp 0,) 2)a Z a = kpp | (ap a) ap a( mod p) a ?= kpa Zp 0(Zp 0,)p 1 ap1= 1ap1 = 1ap1a = a, ap = a p | (ap a), ap a( mod p)? 7.4. P46,Ex4 g, h G gSg1= hSh1 h1gSg1h = S = (h1g)S(h1g)1 h1g N(S) gN(S) = hN(S)gSg1= hSh1 g,hGN(S) GSgSg1|g GG N(S) |O(S)| = G : N(S) ? 8 8.1. P52,Ex1 15 1)hi Hi, hj HjHi, HjG h1 i Hi, h1 j Hj2) (hihj)(h1 i h1 j ) = (hih1 i )(hjh1 j ) = e (h1 i h1 j )1= hihj (h1 i h1 j )1= (h1 j )1(h1 i )1= hjhi hihj= hjhi 2)HiGg G1)gj Hj, j = 1,2,.,n g = g1gi1gigi+1gn 1) g = g1gi1gi+1gngi hi Hi1) (g1gi1gi+1gn)hi= hi(g1gi1gi+1gn)hi hiHi= Hihi= Hi gHi= (g1gi1gi+1gngi)Hi= (g1gi1gi+1gn)Hi = Hi(g1gi1gi+1gn) = Hi(gig1gi1gi+1gn) = Hi(g1gi1gi+1gngi) = Hig HiG 3)GHi1,Hi2,.,Hin (3.1) G = Hi1Hi2Hin1)i, j HiHj= HjHi G = H1H2Hn= Hi1Hi2Hin (3.2)hij, h?ij Hij, j = 1,2,.,n1) (hi1hi2hin)(h?i1h?i2h?in) = (h1h2hn)(h?1h?2h?n) = (h1h?1)(h2h?2)(hnh?n) = (hi1h?i1)(hi2h?i2)(hinh?in) 16 (3.3)g = gi1gi2gin= g? i1g ? i2g ? in gij, g? ij Hij, j = 1,2,.,n1) g = gi1gi2gin= g? i1g ? i2g ? in = g1g2gn= g? 1g ? 2g ? n, gj, g ? j Hj, j = 1,2,.,n c)gj= g? j, j = 1,2,.,n gij= g? ij, j = 1,2,.,n GHi1,Hi2,.,Hin 4)c)gigiGHi (4.1)gi Hig = ee ? ? ? i1 giee ? ? ? ni Gi(g) = gi i (4.2)g, h G, g = g1g2gn, h = h1h2hn, gj, hj Hj b)gh = (g1h1)(g2h2)(gnhn) i(gh) = gihi= i(g)i(h) i iGHi (4.3)g = g1gi1gi+1gn H1Hi1Hi+1Hn g = g1gi1egi+1gn H1Hi1HiHi+1Hn i(g) = e, g Keri g Kerig = g1gi1gigi+1gni(g) = gi= e g = g1gi1egi+1gn= g1gi1gi+1gn H1Hi1Hi+1Hn Keri= H1Hi1Hi+1Hn? 8.2. P53,Ex2 h1h2hn(h1,h2,.,hn)GG (1)g,h G g = (g1,g2,.,gn), h = (h1,h2,.,hn) gh = (g1h1,g2h2,.,gnhn) (gh) = (g1h1)(g2h2)(gnhn) = (g1g2gn)(h1h2hn) = (g)(h) GG (2)g Ggi Hi, i = 1,2,.,ng = g1g2gn g = (g1,g2,.,gn) G(g) = g 17 GG (3)g,h G g = (g1,g2,.,gn), h = (h1,h2,.,hn) (g) = g1g2gn, (h) = h1h2hn (g) = (h)g1g2gn= h1h2hn GGHi c)gi= hii = 1,2,.,n (g1,g2,.,gn) = (h1,h2,.,hn) GG GG? 8.3. P53,Ex3 G = H1H2GH1 1: G H1 g ? h1 1GH1Ker1= H2 H1 = G/H2 H? 1 = G/H2 H1 = H? 1 KleinK = e, a, b, abea2= b2= (ab)2= e H1=, H? 1 =, H2=K = H1H2= H? 1H2 H1 = H? 1 H1?= H? 1 ? 8.4. P53,Ex4 n n = 1 n = 2G =aimi, i = 1,2 (m1, m2) = 1 G |G| | | | | = m1m2 18 a1a2 G(a1a2)m1m2= (a1)m1m2(a2)m1m2= e(a1a2) m1m2(a1a2)ll | m1m2(a1a2)l= (a1)l(a2)l= ee = eeG =(a1)l(a2)l= e (a1)l= (a2)l= em1| l, m2| l lm1, m2l l = m1, m2m1, m2 m1, m2l = m1, m2 = m1m2a1a2G m1m2 n = kn = k + 1H = H,., akH m1mka1akH = G = H Hm1mk mk+1(m1mk, mk+1) = 1 G =(m1mk mk+1)? 9 9.1. P58,Ex1 :Sn2H = e,2= e e HSn Sn1 H = 12s 1,2, ,s iti12t t1, ,tst1,.,ts2= e t1= = ts= 2 = (i1i2)(i3i4)(ij1ij)n 33 (i1i2i3) = (i1i2i3)1= (i3i2i1) 1= (i1i2i3)(i1i2)(i3i4)(ij1ij)(i3i2i1) = (i1i2i3)(i1i2)(i3i4)(i3i2i1)(ij1ij) 19 = (i1i3)(i2i4)(ij1ij)H HSn? 9.2. P58,Ex2 Sn, An, (1)Sn n = 1, 2 n = 3|S3| = 62, 33A3 2S32 H = (1), (i1, i2)(i1i2i3) S3 (i1i2i3)(i1i2)(i1i2i3)1= (i1i3)H HS3 n 5A5HSnSn, An, (1) HAnHAnHAn An K = H An Sn, K H, 1 H An, 1 An 1 KKSn KAnK = (1)K = An K = AnKHAnHAn 2H An? H ? Sn (|An|Sn|) K = (1) K = (1)H , H = = (1)H 2Ex1 SnSn, An, (1) ? 20 10 11 11.1. P71,Ex1 (1)g G, H MgHg1 MG (gHg1)(gHg1) = gHHg1= gHg1, (gHg1)1= (g1)1H1g1) H Me Ge H = eHe1= H g, h G, H Mg (h H) = g (hHh1) = g(hHh1)g1= (gh)H(gh)1= (gh) H (2)HG g GgHg1= H g Gg SH SH= G? 11.2. P71,Ex2 (1)g GgA MgAG m : A gA a ? ga a1,a2 Gga1= ga2a1= a2 x gAa Ax = ga(a) = ga = x A MeGe A = eA = A A M, g, h G g (h A) = g (hA) = g(hA) = (gh)A = (gh) A (2)s SA, a As a = sa AA SA 21 A: x y s SAsx = y AxSAAOxx A = ? xA Ox x A|SA| = |Ox|SAOx : SA Ox s ? sx s1, s2 SAs1x = s2x s1= s2 y Oxs SAy = sxs SA (s) = sx = y x, y A |SA| = |Ox| = |Oy|A Ox|A|Ox| |SA| | m? 12 12.1. P71,Ex1 (1)as,t= (as) s,t s x x kx = aksx lx = altx = ams | m, t | m nx = as,tn= (as,t)n = (2)as a(s,t), at a(s,t) x lx = a(s,t)lm, n ms + nt = (s,t) 22 x = a(s,t)l= (a(s,t)l= (a(ms+nt)l= (as)ml (at)nl =? 12.2. P71,Ex2 G2a Ga2= e, e Ga G a1= a a, b G (ab)1= ab(ab)1= b1a1= baab = ba G ? 12.3. P71,Ex3 H C(G)HGG/H G/HG/H =, a Gx, y G x, yx, yk, lx = (a)k, y = (a)l c1, c2 C(G)x = akc1, y = alc2 xy = akc1alc2= akalc1c2= alakc1c2= alc2akc1= yxG ? 12.4. P72,Ex4 Gp2G1, p, p2 Gp2G Gp2Gp a, b Gap= bp= e0 s pas?= e, bs?= e, eG (ab)p= e (ab)p= e = apbp (ab)p1= ap1bp1 (ab)1= (ab)p1, (a)1= (a)p1, (b)1= (b)p1 (ab)1= a1b1 b1a1= a1b1 (b1a1)1= (a1b1)1 ab = ba G? 12.5. P72,Ex5 G|G| = p1p2ptpi, i = 1,2,.,t 7.6(P43) Gpii = 1,2,.,t Gpiaii = 1,2,.,tH = G H = G, H 23 pk| |H|, pl| |H|, |H|Hpk, pl (pk, pl) = 1Hpkpl akalakalpkplpk, pl akal1, pk, pl, pkpl akal1akal= ea1 k = al akalpk(akal)pk= apk k apk l = apk l = e akalpl(akal)pl= apl ka pl l = apl k = e akalpkplH = a1a2atp1p2ptH = HGHpk| |H|, |H| Hpi, i = 1,2,.,t|H| = p1p2pt