95 5,,,,,,,, ( ) S 3, f1 2 3g f1 2 3g,,, 5.1,,, 1 1 16 G H ' : G! H, '(ab) ='(a)'(b) for 8a b 2 G (5.1), (,, )., 1 1,, ' e 2 G e 0 2 H '(e) =e 0., g 2 G, '(g ;1 )='(g) ;1 : (5.2) 2, 5.1 2 G, H, G H, ' : G! G 0, G 0 H, ' G H,., G 2, ' H 2, ( 1 ). ' : G! H G '(G), H
96 5, ' : G! H '(G) =H,, H G, 37 Z Z m a 2 Z m a a p Z m (p.90 ) p(a + b) =a + b = a + b = p(a)+p(b):, p {p(ab) =p(a)p(b){, p ( 95 ). 97. m, n, Z m Z n 38 S 4 S 3 S n n (4.1)., S 4 4 24, S 3 3 6 ( 74), S 3,1=x 1 ; x., S 4 4 x = a ; c b ; c : a ; d b ; d a, b, c, d a, b, c, d,, x a b c d, ( i.e. a 7! b, b 7! c, c 7! d, b c d a d 7! a), x x x;1 b ; d c ; d : b ; a c ; a = x x ; 1 : 98. x, 4 a, b, c, d 24, x
5.1. 97 98, 6 f (x) f : 7! f, f = f f f (x) = (a) ; (c) (b) ; (c), f (x) =, f (x) =f (f (x)). (a) ; (c) (b) ; (c), : (a) ; (d) (b) ; (d) = y : (a) ; (d) (b) ; (d) = f (y): f, f = f f, 38, f, 98 4 24 3 a $ b b $ c c $ d, S 4,, 1=x, 1 ; x, 1=x, 2 1=x, 1 ; x, 99. +1, ;1,,, 2 f+1 ;1g 39 G 1 T, f(z) =pz + a(jpj =1) (2.3). p, f g(z) =qz + b g g f, '(gf) ='(g)'(f). g(f(z)) = q(pz + a)+b = qpz +(aq + b):, 2, 2, R ' AR ;' B 100, l m, R ' A (l) R' B (m),, l m R ' A (l) R' B (m), R ' A (m) R' B (m)
98 5 100. ABC B, C, E, F, O ABC, AO EF a b a b m n 101. ad ; bc c d c d p q am + bp an + bq, 0, cm + dp cn + dq,, 0 5.2 Z m Z! Z m Z m, mz Z Z m mz Z, Z m = Z=mZ, G, H G H G, Z m eh, g 1 H, g 2 H, ::: He, Hg 1, Hg 2, :::? Z, Z gh = fghjh 2 Hg, Hg = fhgjh 2 Hg, g 2 G ; H G, gh 6= Hg, G = D 3, H = fid S a g, g = R 2 D 3 ; H, D 3 p. 67, RH = fr id R S a g = fr S b g HR = fid R S a Rg = fr S c g:, D 3 C 3 = fid R R 2 g, g D 3 ; C 3,, 2 C 3 D 3 C 3 17 G H, g 2 G, gh = Hg H G, (g 1 H)(g 2 H)=g 1 (Hg 2 )H = g 1 (g 2 H)H = g 1 g 2 H:, g 1 H g 2 H 1, 1 (g 1 g 2 )H,,, g 1 g 2 = g 1 g 2.,, : g = gh = Hg.,, H = eh, gh g ;1 H, H, g 1 H, g 2 H
5.2. 99 18 G, H G G H, g 1 g 2 = g 1 g 2 H G=H G=H, G,, D 3 (p.67),, 1 3 3 4 R, S, R S R R R S S R, Z 2 2, D 3 =C 3 = Z2., = 40 G,, N A G, K G, N K G, N K, G, gh = Hg ghg ;1 = H, H, H H (p.61 ),, (1 ),,, A, A, A, K N, 58(p. 171 ), K f A, f(a), N K G, G=K G=K, 39 G T ' G 2 ', 2 G K, '(f) ='(g) =p, p =1, f, g K p =cos + i sin 6= 1, f g, f = RA, g = R B, f = R A = R B (R ; B RA ) 2 gk.
100 5, R ; B RA, f g K, f = g h, '(f) ='(g)., ' G=K H 1 1 1 1 f f, ' : G=K! H G=K 1, G K,, K ( ),, ' T, 10 ( ) ', G H, K G,, ' H G, G=K = H ' K, G, G=K, k 2 K, f(k) = e, f(gkg ;1 ) = f(g)f(k)f(g) ;1 = f(g)ef(g) ;1 = e gkg ;1 2 K., G, H, K, jgj = jhjjkj., G H, H G 102. (a) D 3 Z 2 (b) Z 3 41 38 ' : S 4! K S 4 ' ',, 2 S 4 =K, jkj =24:6=4., 4, x x a b c d a b c d a b c d b a d c a b c d c d a b a b c d d c b a 4,, D 2, K = D2., S 4 =D 2 = D3 :
5.3. 101 103. f (x) +f (;x) f : R! R F (f (x)) 104. S, C n n, S=C n = S 105., 104 S (R +) 5.3, 3 (p.68),,, 19 S S F (S), S (p.68 ), 2, s k s l = s k+l, s 0 =1, S 1, F (S) 20 R S ( r S r =1 ). R, F (S)=H(R), F (S) S, H(R) R F (S) (, H(R) R F (S) ). s 1 ::: s n, r 1 ::: r m h r 1 ::: r m j r 1 ::: r m i ( r, r =1 )., h a j a n =1i n 42 h a b j ab =1i, Z
102 5 F (a b), a k 1 b l 1 :::a kn b ln H(R), a = bh a = hb (h 2 H(R)), a b 1, a = (ab)b ;1, a b ;1, a b b, F (S)=H(R) b, H(R), a b., b n (n 6= 0) H(R), b, F (S)=H(R), a, b ba ab (i.e.ba = a ;1 (ab)a), ba 2 H(R) 20, H(R) R 106. h a b j a 2 =1 b n =1 aba= b ;1 i? 107. h a b j aba = bab i = h x y j x 2 = y 3 i : 5.4, 21 G, X 57 ) X G, G, X (p. T : G! Tr(X): T g, x 2 X T g (x) T g gx G, X, G! Tr(X), S 3,, 3 5.1, O, a, b, c D 3 (O), a, b, c 3 ( ),, D 3 6
5.4. 103 5.1: D 3 22 x 2 X O(x), x O(x) =ft g (x) j g 2 Gg: O(x), 3 1 3 6,, 1, 5.1, 1 O 5.1 A = A 1, fa 1 A 2 A 3 g A 2 A 3,, x O(x), y 2 O(x), h 2 G y = hx, O(y) =fgyjg 2 Gg = fghxjg 2 Gg,, h 2 G, G = fghjg 2 Gg = G 2,, 2 O(x) O(y) z, O(x) =O(z) =O(y),, S, 1 ( 0),, 108. D 3,, 1 x, f0 1 ;1 2 ;2 i p 3 ;i p 3 4 ;4 2+2i p 3 2 ; 2i p 3 ;2 + 2i p 3 ;2 ; 2i p 3g D 3
104 5 5.2: S 43, ( 74) 1,.,, 0 2 1, R 2 0, 1 ( x 1 ), 1, R 1 R, 1 F 1 F 2 F 3 F 4 F 5 F 6 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1, R [1, 3 0, 1, 1,, 3 D 3 109., 3 ( ) 1,, 6 110. F i 2, C [1,, 2, 2 2 3, 6 5.5
5.5. 105 44 Q, G, Q Q (i.e. Q ), G, Q G G 2 (, 5.2 AA 0 ) 180 6. 2 (, BB 0 ) 180 3. 90 6. (, CC 0 ) 120 8., G 24 5.3: F G,,,, F, 6, 1 f 2 F, f 4, 3 6 2, G 24 111. E V G, 6 8 12 G, 24 ( ) 24
106 5,,, G 5.4, 4, 3, 2 D, M,, T 23 a b c 5.4:, G 112. F, E, V, D, M, a) b), 112 45 (f e) G, f 2 F Q, e 2 E, F E ( F E ), 6 12 = 72, 3 e f ( 5.5a), e f 1 ( 5.5b) e f ( 5:5c), 3 1 2, ( ) 2 (f 1 e 1 ), (f 2 e 2 ), f 1 f 2,, 4 e 1 e 2 2, G, F E, 3, 5.5
5.5. 107 a b c 5.5: { 113. G, (a) V F ( { ) (b) D F ( { ) (c) E E( )., 24 X G, x 2 X, x G st(x) St(x) =fg 2 G j T g (x) =xg:, G, g h 2 St(x), T gh (x) =T g (T h (x)) = T g (x) =x T g ;1(x) =T g ;1 (x) =x:, G, 5.1 O,, A, 2 ( ), B, ( ),, 44,, jo(x)jj St(x)j = jgj: (5.3) 5.3, G H =St(x) G = g 1 H [ g 2 H [[g k H:, x, x, h 2 H, T gh (x) = T g (T h (x)) = T g (x), g, k 2 G x, g k H, k = gh h 2 G, T h (x) =T g ;1 k(x) =T ;1 g T k (x) =x
108 5, h = g ;1 k 2 St(x)., x G H G= St(x), 5.3 5.3,,,, G,, G X, O(x) x 2 X, 2 x, y 2O(x), St(x) St(y),, y = T g (x) g 2 G g St(x) St(y), h 2 St(x), T ghg ;1(y) =T g (T h (T g ;1 (y))) = T g (T h (x)) = T g (x) =y:, g St(x)g ;1 St(y). x y St(y) g St(x)g ;1, g St(x)g ;1 = St(y)., St(x) St(y) 46 p n,, p, 2,, (n p ) C p,, 2, 1 p, p 1,,, 1 n n p ; n, p, (n p ; 1)=p,,, (n p ; n)=p + n p, n p ; n p, 4.6 114. 46 p 114, 46,, Burnside g 2 G, X, T g, T g (x) =x x 2 M N(x)
5.5. 109 115. 0, 1, 8, (180 ), 6, 9, 2, 3, 4, 5, 7, 180, 180 5 11 (Burnside ) r, G, r = 1 jgj X g2g N(g):, x 2 M, P g2g N(g), g x, g,, j St(x)j O(x),,,, j St(x)jjO(x)j,, G jgj, jgj, jgj ( ) Burnside, 46, n p, n, r =(n p +(p;1)n)=p, 47 7 3 2,,,,, 7 3 X, X D 10 Burnside, D 10 X, X ; 10 3 =120 (a) ( 5.6),, (b),, 24 =8, 1, 1, 4 1 (b) 5, Burnside r =(120+58)=20 = 8.
110 5 5.6: 3-7 116. 47, 6, 4 117., 1 6 (2, ). 118. 2 (i.e. 2 ) (a) (b), 117 119. 15 (,2 ). 5.6, Burnside, 3, 3 m, n, k, m, n, k 0 7, m + n + k =7, (m n k), m, n, k, (4!), m n k, 3, (0 0 7), (0 1 6), (0 2 5), (0 3 4), (1 1 5), (1 2 4), (1 3 3), (2 2 3)., 3 (m n k), 2 2 ( ), 3 2 3, 2
5.6. 111 m, n, k 25 G X, X N ',,, ', N, 3 3 ( m ),,, 2, 5.7 2,, m, 26 ' : X! N, 5.7: 120. 4 6 M, M, M,, M, 2, 2., 121. M (a) (b) H M, H H,,, H A, A A A (p.25
112 5 ), A r, r f(r), 2 D 3 ( 5.1 ) O, OM, r,, r 1 cos '., cos ' D 3, D 3 ' 7! ;' ' 7! ' +120, cos 3', (r cos 3'),, cos 3' = b r = c, c>0 jbj1, 1 3, 6 5.7, (p.5 ). 2, 27 3.1b 27 2 28 G, G, A, A, A, 1, 2,, A, A 122., 123.
5.7. 113, 29 G, 2 1 F, x 2 F ( ) F 2, 2, F ( ),, F, ( 27), 27, 124. C n D n,,, 3, p3m1 p1, 2 a, b 5.8a, p1 b a a b c d 5.8:, a b 5.8b, p1. 2, 1 1 2, b;a b 2 2, a +2b 2a +3b 125. ka + lb, ma + nb 2,, jkn ; lmj =1, k, l, m 1,
114 5 p1, p1, p1,, ( 5.8c)., p1,,,, 5.8d p1,, 1., p1 (i.e. 2 ) 1, G, g 2 G, g l, G, G,, G l 2,, 180, R ' 2 G, ' 6= 180,,, T a 2 GT a R ' T a R ;' ( 58 ). 2 180,, G, l l,, G 126. G, l, l, 2 l, 1, 1, G, G, G, 2 p1 2. G p1, 127., 2
5.7. 115, p1 G, H G (H p1 )., H G, ( 8 ). 3. G, G=H, 10 C n, D n (n =1 2 3 4 6) 1, ( 130). G=H G G H G, g 1 g k G=H,, = k[ i=1 H,, g i,, ( ) ( ) G H (i.e. G=H ) G=H, 48 p3m1, H G g i 5.9a, p3m1,, 3 3 ( 5.9a, 3.1b )., MNB ( ), A B D, ABCD 2 ;! ;! AB, AD G H, S ABCD : S MNB =6:1 ABCD, 6 (, 6 )., H, BQCLDM, 6 G=H, H (` ' ), g h 1 g h 2 (h 1 h 2 2 H)
116 5 5.9: p3m1, h 1 h 2,, 6, G=H 6 5.9b, 1 6,, gh,, g 2 G 1 $ 6, 2 $ 5, 3 $ 4. G=H D 3,, 3 3, G=H = D3., G D 3 p3m1=p1 p3m1 p1 128. G=K, G, K H ;! AC ;! AR ( 5.9a ).,, C n, D n 1 M p1?, 2 a, b,, 2 p1 (p.123 ): H T a, T b, K T c, T d, LHL ;1 = K, L L(a) =c, L(b) =d 30 2,
5.7. 117, 17 12 (Fedorov{Schoniess),, 17 1 17,,,, 1., 2, 3, 4, 6 2. G + G G ( ), G +, (i.e. G=G + ) 1 2 3. G + = G(i.e. G ), G, p1, p2, p3, p4, p6, G 4. G + 6= G, G G + G n G + f, G +, 5 (f,, ), (a) G + = p1, G = pm, pg, cm. (b) G + = p2, G = pmm, pmg, pgg, cmm. (c) G + = p3, G = p31m p3m1. (d) G + = p4, G = p4m p4g. (e) G + = p6, G = p6m. ( ) : : (,, 4,,, 2, 3, 4, 6 ). : : 1 12, 13 17,, p1, p2, pm, pg, pmm, pmg, pgg, cm, cmm,,
118 5 p1 T 1, T 2 T 1 T 2 = T 2 T 1 R 1, R 2, R 3 p2 R 2 1 = R2 2 = R3 3 =id, (R 1 R 2 R 3 ) 2 =id pm S 1, S 2 T S 1 T = TS 1, S 2 T = TS 2, S 2 1 = S2 2 =id pg U 1, U 2 U 2 1 = U 2 2 cm S U S 2 = id, SU 2 = U 2 S S 1, S 2, S 3, S 4 pmm S 2 1 = S2 2 = S2 3 = S2 4 = id, (S 1 S 2 ) 2 =(S 2 S 3 ) 2 =(S 3 S 4 ) 2 =(S 4 S 1 ) 2 =id S R 1, R 2 pmg S 2 = R 2 1 = R2 2 =id, R 1 SR 1 = R 2 SR 2 pgg U 1, U 2 (U 1 U 2 ) 2 =(U ;1 1 U 2 ) 2 =id
5.7. 119 S 1, S 2 R cmm S 2 1 = S 2 2 = R 2 =id, (S 1 S 2 ) 2 =(S 1 RS 2 R) 2 =id p4 R 90 R 1 R 2 = R 4 1 =(R 1 R) 4 =id S 1, p4m S 2, S 3 S 2 1 = S 2 2 = S 2 3 =id, (S 1 S 2 ) 2 =(S 2 S 3 ) 4 =(S 3 S 1 ) 4 =id p4g S 90 R S 2 = R 4 =(R ;1 SRS) 2 =id p3 3 120 R 1, R 2, R 3 R 3 1 = R 3 2 = R 3 3 = R 1 R 2 R 3 =id p31m R 120 R R 3 = S 2 =(R ;1 SRS) 3 =id S 1, S 2, S 3 p3m1 S 2 1 = S2 2 = S2 3 =id, (S 1 S 2 ) 3 =(S 2 S 3 ) 3 =(S 3 S 1 ) 3 =id p6 R 120 R 1 R 2 = R 3 1 =(R 1 R) 6 =id (30 60 90 ) p6m S 1, S 2, S 3 S 2 1 = S 2 2 = S 2 3 =id, (S 1 S 2 ) 2 =(S 2 S 3 ) 3 =(S 3 S 1 ) 6 =id
120 5 129. 4 5.8d 130., 3 (p. 115) 131.