2011 (2011/02/08)

1 7 1.1.................................... 7 1.2..................................... 8 1.3.................................. 9 1.4.................................. 10 1.5.................................. 11 1.6..................................... 12 1.7............................... 12 1.7.1................................. 12 1.7.2........................... 13 1.8..................................... 14 1.8.1................................. 14 1.8.2................................. 15 1.9............................... 17 2 19 2.1.................................... 19 2.2.................................. 22 2.3.................................... 23 2.4.................................... 24 3 27 3.1.................................... 27 3.2............................. 29 3.3............................ 30 3.4................................ 31 4 35 4.1................................. 35 5 39 5.1.................................... 39 5.2.................................... 40 3

4 6 43 6.1................................ 43 6.2............................. 45 6.2.1....................... 46 6.2.2........................... 47 6.2.3................... 48

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Chapter 1 1.1 A f : A A A A f(a, b) ab a + b (a, b) ab a, b, c A (ab)c = a(bc) A e A a A ae = ea = a A 1 A 1 ( (a, b) a + b 0 A 0 ) A a ab = ba = 1 A b a A b a a 1 ( (a, b) a + b a ) A A (a, b) ab a, b A ab = ba (a, b) a + b A ( ) A A A A 7

8 CHAPTER 1. 1.1.1. G a, b, a i (1) (a 1 ) 1 = a (2) (ab) 1 = b 1 a 1 (a 1 a l ) 1 = a l 1 a 1 1 1.2 G G H G H G H (1) x, y H xy H x H x 1 H (2) x, y H xy 1 H (3) x, y H x 1 y H 1.2.1. G G G {1 G } G G {1 G } 1 H G H G H < G H G H G A, B G AB = {ab a A, b B}, A 1 = {a 1 a A} A(BC) = (AB)C, (AB) 1 = B 1 A 1 (4) HH H, H 1 H (5) HH 1 H (6) H 1 H H H G H = HH = H 1 = HH 1 = H 1 H A = {a} {a}b ab Ab ab = {ab b B}, Ab = {ab a A}

1.3. 9 1.2.2. H, K G H K G {H λ λ Λ} G λ Λ H λ G G Z(G) = {g G x G xg = gx} G (center) G Z(G) = G 1.2.3. G Z(G) G G S G S G S S S S = {s 1,, s l } s 1,, s l 1.2.4. G H, K HK G HK = KH 1.3 G H a, b G a 1 b H a b G a G ah = {ah h H} G H a (left coset) G/H {a λ H λ Λ} G = λ Λ a λ H G H G/H G = ah a G/H G H ( ) G : H H G (index) a G ah = H G 1.3.1 ( ). G H G = G : H H H G

10 CHAPTER 1. G a a n = 1 n n a o(a) a = {1, a, a 2,, a o(a) 1 } G o(a) G a a r ab 1 H a r b Ha = {ha h H} G H a H \ G {a λ λ Λ} G H {a λ 1 λ Λ} ( ) ( ) 1.4 a G ah = Ha H G (normal subgroup) H G H G H G H G H (coset) G 1 G G G 1.4.1. G H (1) H G ( g G gh = Hg) (2) h H, g G ghg 1 H (3) g G ghg 1 H (4) g G ghg 1 = H. (1) = (2). h H, g G gh = Hg gh gh = Hg h H gh = h g ghg 1 = h H (2) = (1). g G gh = Hg x gh h H x = gh ghg 1 H x = gh = (ghg 1 )g Hg

1.5. 11 gh Hg y Hg h H y = hg g 1 hg = g 1 h(g 1 ) 1 H y = hg = g(g 1 hg) gh Hg gh gh = Hg (2) (3) (4) = (3) (3) = (4) g G ghg 1 H g 1 G (3) g 1 Hg H g g 1 H ghg 1 (4) 1.4.2. H, K G H K G {H λ λ Λ} G λ Λ H λ G 1.4.3. H G N G HN G H, K G HK G 1.4.4. H G N G H N H. 1.5 G H, K G a, b G a b HaK = HbK HaK = {hak h H, k K} a HaK G (H, K) (double coset) H \ G/K G = HaK a H\G/K HaK = h H hak HaK K HaK H 1.5.1. G H, K a G HaK H : H aka 1 K HaK = H : H aka 1 K. f : H/(H aka 1 ) {hak h H} f(h(h aka 1 )) = hak h(h aka 1 ) = h (H aka 1 ) k K h = h(aka 1 ) h ak = h(aka 1 )ak = hak f

12 CHAPTER 1. f f hak = h ak, h, h H a 1 (h ) 1 ha K, (h ) 1 h H aka 1 h(h aka 1 ) = h (H aka 1 ) f H G HaK HaK = ahk HK 1.6 G H G/H H G/H H G G/H (ah)(bh) = (ab)h H = 1H a 1 H ah G/H G H (factor group) N H G N G G/N H/N = {hn h H} G/N h, h H (hn)(h N) 1 = (hn)((h ) 1 N) = (h(h ) 1 )N H/N N H H/N H/N 1.7 1.7.1 X x X x 1 X 1 = {x 1 x X} (x 1 ) 1 = x X X 1 (word) xx 1 X X 1 0 1 1 X (free group) F (X) 1.7.1 ( ). X = {a} F (X) = {a i i Z} Z

1.7. 13 1.7.2 X F (X) R F (X) R F (X) N R F (X) N N F (X)/N X R X R F (X) R 1 X R R r r = 1 1.7.2 ( ). a a n = 1 n ( ) ( (cyclic group) ) n C n C n = {1, a, a 2,, a n 1 } Z n Z/nZ 1.7.3 (). x, y x n = y 2 = 1, yx = x n 1 y 2n (dihedral group) 2n D 2n D 2n = {1, x, x 2,, x n 1, y, xy, x 2 y,, x n 1 y} 1.7.4. G = a a n = 1 (1) a m = 1 n m (2) o(a m ) = n/ gcd(m, n) gcd(m, n) = 1 o(a m ) = o(a) a m = a. (1) (2) d = gcd(m, n) m = dm, n = dn (a m ) n = a m n = 1 o(a m ) n a m = o(a m ) n x, y xm + yn = d a d = a xm+yn = (a m ) x (a n ) y = (a m ) x a m a d a m o(a d ) = n n = a d a m = o(a m ) o(a m ) = a m = n = n/d

14 CHAPTER 1. 1.7.5. G n d n {x G x d = 1} = d 1.7.6. N G g G G/N gn o(gn) o(g) 1.8 1.8.1 K n M n (K) M n (K) ( M n (K) K n ) 1.8.1 ( GL n (K)). K K n 0 K n K n (general linear group) GL n (K) K q F q F q GL n (K) GL n (q) 1.8.2. GL 2 (2) F 2 = {0, 1} 1.8.3. GL n (q) 1.8.4 ( SL n (K)). K n 1 GL n (K) K n (special linear group) SL n (K) SL n (F q ) SL n (q) 1.8.5. SL 2 (3) F 2 = {0, 1, 2} 1.8.6 ( O(n)). T M n (R) (orthogonal matrix) t T T = T t T = E (E ) t T = T 1 n GL n (R) n (orthogonal group) O(n) 1.8.7. O(n) GL n (R). 1.8.8 ( U(n)). U M n (C) (unitary matrix) U U = UU = E U = U 1 U = t U n GL n (C) n (unitary group) U(n) 1.8.9. U(n) GL n (C).

1.8. 15 1.8.2 X X X X Sym(X) Sym(X) X X X σ ( ) x σ = σ(x) X = n < X = {1, 2,, n} Sym(X) S n n S n n n 1 1, 2,, n ( ) 1 2 3 σ = 3 2 1 ( ) ( ) 1 2 3 1 2 3 = 3 2 1 2 1 3 ( 1 2 ) 3 2 3 1 1 2 ( ) 1 1 2 3 = 2 3 1 ( ) 2 3 1 = 1 2 3 ( 1 2 ) 3 3 1 2 1 σ a 1, a 2,, a l a 1 σ a 2 σ a 3 σ σ a l σ a 1 (a 1 a 2 a l 1 a l ) ( (a 1, a 2,, a l 1, a l ) ) l l l 2 1 ( ) ( ) ( ) 1 2 3 1 2 3 4 = (1 2 3), = (1 2) 2 3 1 2 1 3 4 1.8.10. n

16 CHAPTER 1. ( ) 1 2 3 4 5 6 7 8 9 1.8.11. 3 6 5 8 1 2 7 9 4 1.8.12. n 1.8.13. (1 2 l) = (1 l)(1 l 1) (1 3)(1 2) σ ( ) σ [l 1, l 2,, l r ], l 1 l 2, l r 1 l i = 1 l i σ = [3, 2, 1] ( [3, 2]) ( ) 1 2 3 4 5 6 = (1 2 3)(4 5)(6) 2 3 1 5 4 6 1.8.14. σ [l 1, l 2,, l r ] σ o(σ) l 1, l 2,, l r 1.8.15. (1) σ, τ σ τστ 1 (2) σ σ τ σ = τστ 1 n σ ( ) σ 1 1 sgn(σ) S n n A n n 5 n A n 1 A n 1 (simple group) 1.8.16. (1 2 3 4 5), (1 2 3 4)(5 6) 1.8.17. A 4 1 1980 20

1.9. 17 1.9 M U(M) M U(M) M M 1.9.1. K n M n (K) GL n (K) 1.9.2. X X Sym(X) 1.9.3. R U(R) R 1.9.4. ( ) K U(K) = K {1 K } K 1.9.5. Z U(Z) = {1, 1} 1.9.6. 1.9.7. G 1 G G 1.9.8. G g g 2 = 1 G 1.9.9. G Z(G) G/Z(G) G ( G = Z(G) ) ( ) a b 0 a b 1.9.10. K M = GL c d 2 (K) M = c d 0 0 0 1 M GL 3 (K) GL 2 (K) GL 3 (K) GL 2 (K) GL 3 (K) ( m n GL m (K) GL n (K) ) 1.9.11. X Sym(X) Y X Sym(Y ) X Y Sym(X) Sym(Y ) Sym(X) Sym(Y ) Sym(X) ( m n S m S n ) 1.9.12. 8 G = D 8 = x, y x 4 = y 2 = 1, yx = x 3 y H = y G H (H, H)

18 CHAPTER 1. 1.9.13. Q 8 = {±1, ±i, ±, j, ±k} i 2 = j 2 = k 2 = 1, ij = ji = k, jk = kj = i, ki = ik = j 1 1 Q 8 Q 8 8 D 8 Q 8 D 8 1.9.14. V = x, y xy = yx, x 2 = y 2 = 1 V (V ) 1.9.15. GL n (C)

Chapter 2 2.1 G, H f : G H (group homomorphism) a, b G f(ab) = f(a)f(b) f(a + b) = f(a) + f(b) 2.1.1. G H f : G H 2.1.2. K GL n (K) det : GL n (K) K det(ab) = det(a) det(b) 2.1.3. n σ sgn(σ) S n {1, 1} 2.1.4. K V, W K- f : V W f(v + v ) = f(v) + f(v ) 2.1.5. R R = R {0} exp : R R exp(x + y) = exp(x) exp(y) 2.1.6. G H ι : H G, ι(h) = h 2.1.7. f : G H, g : H K g f : G K 19

20 CHAPTER 2. 2.1.8. f : G H (1) f(1 G ) = 1 H (2) a G f(a 1 ) = f(a) 1. (1) f(1 G ) = f(1 G 1 G ) = f(1 G )f(1 G ) f(1 G ) 1 1 H = f(1 G ) (2) 1 H = f(1 G ) = f(aa 1 ) = f(a)f(a 1 ) f(a) 1 f(a) 1 = f(a 1 ) f : G H Kerf = {a G f(a) = 1 H } f (kernel) f (image) Imf = {f(a) a G} 2.1.9. f : Z/4Z Z/4Z f(a + 4Z) = 2a + 4Z f f 2.1.10. f : G H (1) Kerf G (2) Imf H. (1) f(1 G ) = 1 H 1 G Kerf Kerf x, y Kerf f(xy 1 ) = f(x)f(y) 1 = 1 H xy 1 Kerf Kerf G x Kerf, g G f(gxg 1 ) = f(g)f(x)f(g) 1 = f(g)f(g) 1 = 1 H gxg 1 Kerf Kerf G (2) 1 H = f(1 G ) Imf Imf a, b Imf x, y G a = f(x), b = f(y) Imf H ab 1 = f(x)f(y) 1 = f(xy 1 ) Imf 2.1.11. S n sgn sgn A n (Ker(sgn) = A n ) f : G H f (monomorphism) f (epimorphism) f (isomorphism)f f : G H G H G H G = H

2.1. 21 G G G G = G f : G H f f 1 : H G G = H H = G f : G H g : H K g f : G K G = H H = K G = K G = {a, b}, H = {x, y} a b a a b b b a x y x x y y y x G H f : G H f(a) = x, f(b) = y f 2.1.12. f : G H f Kerf = 1. f 1 G Kerf x Kerf f(x) = 1 H = f(1 G ) f x = 1 G Kerf = 1 Kerf = 1 f(x) = f(y) 1 = f(x)f(y) 1 = f(xy 1 ) xy 1 Kerf = {1 G } x = y f 2.1.13. N G f : G G/N f(g) = gn Kerf = N 2.1.14. f : G H (1) A G f(a) H (2) B H f 1 (B) G (3) B H f 1 (B) G (4) f A G f(a) H. (1) x, y A f(x)f(y) 1 = f(xy 1 ) f(a) f(a) H (2) s, t f 1 (B) f(s), f(t) B f(st 1 ) = f(s)f(t) 1 B st 1 f 1 (B) (3) f 1 (B) G (2) g G, s f 1 (B) f(s) B H f(gsg 1 ) = f(g)f(s)f(g) 1 f(g)bf(g) 1 B

22 CHAPTER 2. gsg 1 f 1 (B) (4) f(a) B (1) x f(a), h H a A x = f(a) f g G h = f(g) hxh 1 = f(g)f(a)f(g) 1 = f(gag 1 ) f(gag 1 ) f(a) 2.1.15. f : G H A G f(a) H 2.2 2.2.1. f : G H N Kerf G f : G/N H f(gn) = f(g). f an = bn n N b = an n N Kerf f f f(b) = f(an) = f(a)f(n) = f(a)1 H = f(a) f((an)(bn)) = f((ab)n) = f(ab) = f(a)f(b) = f(an)f(bn) 2.2.2 ( ). f : G H f : G/Kerf Imf f(g(kerf)) = f(g) G/Kerf = Imf f G/Kerf = H. N = Kerf 2.2.1 f : G/N H (gn f(g)) Imf = Imf f : G/Kerf Imf Imf f an Kerf f(an) = 1 H f(a) = 1 H a Kerf an = 1 G N Kerf = 1 f f

2.3. 23 2.2.3. f : Z/4Z Z/4Z f(a + 4Z) = 2a + 4Z f ( 2.1.9 ) a + 4Z a f Kerf = {0, 2} = 2Z/4Z, Imf = {0, 2} = 2Z/4Z (Z/4Z)/(2Z/4Z) = 2Z/4Z 2.3 ( 2.2.2) 2.3.1 (). H G N G HN/N = H/(H N). f : H HN/N f(h) = hn x HN h H n N x = hn xn = hnn = hn = f(h) f f(h) = hn = 1N h N Kerf = H N HN/N = H/(H N) 2.3.2. G = Z, H = 4Z, N = 6Z H+N = 4Z+6Z = 2Z, H N = 12Z 2Z/6Z = 4Z/12Z f : 2Z/6Z 4Z/12Z, f(a + 6Z) = 2a + 12Z 2.3.3 (). N H G N H G/H = (G/N)/(H/N). f : G G/H Kerf = H N g : G/N G/H, g(xn) = xh Kerg = H/N G/H = (G/N)/(H/N)

24 CHAPTER 2. 2.3.4. 2.2.3 (Z/4Z)/(2Z/4Z) = 2Z/4Z (Z/4Z)/(2Z/4Z) = Z/2Z Z/2Z = 2Z/4Z f : Z/2Z 2Z/4Z, f(a + 2Z) = 2a + 4Z N G p : G G/N S G N T G/N H S H/N = p(h) T X T p 1 (X) S 2.3.5. N G G N G/N 2.3.6. 2.3.5 2.4 G G G G (automorphism) G G Aut(G) 2.4.1. GF (2) 2 V V = {(0, 0), (0, 1), (1, 0), (1, 1)} V {(0, 0)} V Aut(V ) = S 3 g G f g : G G, x gxg 1 f g G G (inner automorphism) f : G Aut(G), g f g f Aut(G) Inn(G) f g Kerf f g = id G x G gxg 1 = x Kerf G Z(G)

2.4. 25 2.4.2. Inn(G) = G/Z(G) 2.4.3. Inn(G) Aut(G). σ Inn(G), τ Aut(G) g G x G σ(x) = f g (x) = gxg 1 f g y G (τστ 1 )(y) = τ(gτ 1 (y)g 1 ) = τ(g)yτ(g) 1 τστ 1 = f τ(g) Inn(G) Inn(G) Aut(G) Aut(G)/Inn(G) G Out(G) G H G σ Inn(G) σ(h) = H G H G (characteristic subgroup) σ Aut(G) σ(h) = H H G, K H K G 2.4.4. H G K H K G. k K, g G gkg 1 = f g (k) H G f g Aut(H) K H gkg 1 K 2.4.5. H G K H K G. σ Aut(G) σ H Aut(H) σ(k) = (σ H )(K) = K 2.4.6. Z(G) G 2.4.7. 2.4.8. G, H f : G H f f 1 : H G f 1 2.4.9. G G G 2.4.10. G, H f : G H g G o(g) < o(f(g)) < o(f(g)) o(g) ( f o(g) = o(f(g)) )

26 CHAPTER 2. 2.4.11. G a, b G [a, b] = aba 1 b 1 a b (commutator) G G (derived subgroup) D(G) [G, G] (1) [a, b] = 1 ab = ba (2) D(G) G (3) N G G/N D(G) N (4) M G, N G G/M, G/N G/(M N) 2.4.12. G D 0 (G) = G D n+1 (G) = D(D n (G)) D n (G) n- n D n (G) = 1 G 1 (solvable group) G ( ) 2.4.13. n- D n (G) G 2.4.14. G A, B G A B = 1 a A b B ab = ba 2.4.15. G, A ϕ : A Aut(G) G A g, h G, a, b A (g, a)(h, b) = (gϕ(a)(h), ab) G A ( G A (semidirect product) G A ) 2.4.16. 2 2.4.17. n ϕ(n) ϕ(n) n n 1 5 A 5 5

Chapter 3 3.1 X G f : G X X f(g, x) gx f f G X () (A1) x X 1 G x = x (A2) x X, g, h G (gh)x = g(hx) f G X ( ) X G- xg gh g h gx xg g x x g 3.1.1. K GL n (K) ( 1.8.1 ) K n K n K n 3.1.2. K GL n (K) M n (K) M M n (K) P GL n (K) P M = P MP 1, M P = P 1 MP (A1) (A2) (P Q) M = (P Q)M(P Q) 1 = P QMQ 1 P 1 = P (QMQ 1 ) = P ( Q M) 27

28 CHAPTER 3. P M = P MP 1 M (P Q) = (P Q) 1 M(P Q) = Q 1 P 1 MP Q = (P 1 MP ) Q = (M P ) Q M P = P 1 MP 3.1.3 ( ). G X g G x X gx = x 3.1.4. S R n O(n) n ( 1.8.6 ) M S T O(n) T M = T MT 1 O(n) S f : G X X G X g G f g : X X, f g (x) = gx f g f g 1 = f g 1 f g = id G f g f g Sym(X) F : G Sym(X), F (g) = f g g, h G x X F (gh)(x) = f gh (x) = (gh)x = g(hx) = f g (f h (x)) = F (g)(f (h)(x)) = (F (g) F (h))(x) = (F (g)f (h))(x) F (gh) = F (g)f (h) F F : G Sym(X) f : G X X gx = f(g, x) = F (g)(x) G X G X G Sym(X) G Sym(X) G X (permutation representation) G X f : G Sym(X) G X (faithful) f 3.1.5. G X (1) G X (2) g G x X gx = x g = 1 G X f : G Sym(X) G Sym(X) Sym(X) X (permutation group) S n n X G X G X f : G Sym(X) G/Kerf

3.2. 29 3.2 G X x, y X x y g G y = gx 3.2.1. X X G (orbit) G- x X ( ) C x C x = {gx g G} G X C x Gx ( g x, x g G x, x G ) g G Gx G G g, h G gx = hx gx = hx x = 1x = (g 1 g)x = g 1 (gx) = g 1 (hx) = (g 1 h)x G x = {g G gx = x} x G G x G (G x Gx ) 3.2.2. G x G 3.2.3. S 4 G = (1 2), (3 4) = {( ), (1 2), (3 4), (1 2)(3 4)} G X = {1, 2, 3, 4} X G {1, 2}, {3, 4} X X = {1, 2} {3, 4} G 1 = G 2 = {( ), (3 4)}, G 3 = G 4 = {( ), (1 2)} gx = hx x = (g 1 h)x g 1 h G x gg x = hg x 3.2.4. G X x X f : G/G x Gx, f(gg x ) = gx G : G x < G : G x = Gx

30 CHAPTER 3.. gg x = hg x gx = hx f Gx g G gx f f(gg x ) = f(hg x ) gx = hx gg x = hg x f X G (transitive) G X x, y X g G y = gx X G- (transitive G-set) G X x X Gx G Gx f : G X X f G Gx Im(f G Gx ) Gx f : G Gx Gx G Gx G- G- G- 3.2.5 ( G- ). G G G (g, x) gx G G- ( ) G- G X Y X G y Y g G gy Y y Y Y Y G- G Y 3.3 G H H G G/H G/H = {ah a G} g(ah) = (ga)h ah, bh G/H bh = (ba 1 )(ah) G- X Y X Y G- f : X Y x X, g G gf(x) = f(gx)

3.4. 31 f G- 1 X Y G G X X id G f G Y Y f 3.3.1. G X x X f : G/G x X f(ag x ) = ax f G- G- G-. ag x = bg x ax = bx f ax = bx ag x = bg x f G f f g G, ag x G/G x gf(ag x ) = g(ax) = (ga)x = f((ga)g x ) = f(g(ag x )) f G- 3.4 G G G g G x G g x = gxg 1 G G- G G x g = g 1 xg G G ( ) x G g G x x = g x = gxg 1 xg = gx G x C G (x) G x (centralizer) C G (x) = {g G xg = gx} S G C G (S) = s S C G (s) 1 A B

32 CHAPTER 3. G S C G (G) G Z(G) x G G x = { g x g G} = {gxg 1 g G} G x (conjugacy class) 3.4.1. G x G x G x G : C G (x). 3.2.4 G N N G N G G G G {x 1 = 1, x 2,, x r } G = G x 1 G x 2 G x r G = G x 1 + G x 2 + + G x r G (class equation) C G (1) = G G x 1 = G 1 = 1 G x i G : C G (x i ) G 3.4.2. G x = 1 x Z(G) 3.4.3. 3 S 3 C 1 = {1}, C 2 = {(1 2 3), (1 3 2)}, C 3 = {(1 2), (1 3), (2 3)} 6 = 1 + 2 + 3 S 3 C 1 C 2 C 1 C 3 S 3 6 C 1 C 2 C 1 C 2 1 Z(S 3 ) = 1

3.4. 33 3.4.4. p G p n N 1 G N Z(G) 1. N G G N G C 1 = {1 G }, C 2,, C l x i C i G x i = G : C G (x i ) G p n i G x 1 = 1 l l N = p n i = 1 + p n i i=1 N G N 1 N p p n 1 = 1 i n i > 0 p i 1 n i = 0 1 x i Z(G) N p p- 3.4.5. p- 1. 3.4.4 G = N A G g G i=2 g A = gag 1 = {gag 1 a A} G G ( ) 2 G A A (normalizer) N G (A) N G (A) = {g G g A = A} = {g G gag 1 = A} = {g G ga = Ag} A = g A 3.4.6. N G (A) G H G H N G (H) G. N G (A) 2 G G G H G h H hh = H = Hh H N G (H) 3.4.7. H G g H G. x, y g H a, b H x = gag 1, y = gbg 1 xy 1 = (gag 1 )(gbg 1 ) 1 = gag 1 gb 1 g 1 = g(ab 1 )g 1 ab 1 H xy 1 g H g H H G G H G N G (H) = G H G H N G (H) H N G (H) 3.4.8. H G H G G : N G (H). H G H 3.2.4

34 CHAPTER 3. 3.4.9. n Z/nZ G (1) g G a + nz Z/nZ g(a + nz) ga + nz G Z/nZ (2) n = 5, 6, 8 (1) 3.4.10. K V = K n K n GL n (K) V 1 0 e e GL n (K) 3.4.11. 8 G = D 8 = x, y x 4 = y 2 = 1, yx = x 3 y H = y D 8 N G (H) 3.4.12. 4 A 4 (1) A 4 (2) A 4 (3) A 4 12 3.4.13. Q 8 3.4.14. p p 2 3.4.15. G n a i, b i {1,, n}, a i b i (i = 1, 2) g G g(a 1 ) = a 2, g(b 1 ) = b 2 G 2 ( t ) G n G 1 = {σ G σ(1) = 1} G 1 {2, 3,, n} G 2 G 1 {2, 3,, n}

Chapter 4 4.1 (Sylow) p 4.1.1. G p G G p. G G = p G p G > p 1 g G n = o(g) p n o(g n/p ) = p p n H = g 1 < H < G p H p G/H G/H < G p G/H G/H p a a l = 1 p l a l = 1 p l p o(a) o(a o(a)/p ) = p 4.1.2. G ( 1) p G Z(G) p Z(G) 1. G = G 1 + G x 1 + + G x r G x i = G : C G (x i ) C G (x i ) G G x i p G = G : 1 p C G (1) = G G 1 = 1 p p G x i = 1 i C G (x i ) = G x i Z(G) p p- G p- p- G G = p a q, p q G p a G p- p- G p- 35

36 CHAPTER 4. 4.1.3 ( (1)). p r G G p r G p-. G G = 1 G > 1 p r G r = 0 r > 0 G H G : H p H p r G p 4.1.2 Z(G) p 4.1.1 Z(G) p a N = a a Z(G) N G N = p G/N p r 1 G/N G/N p r 1 H/N H = p r G p r G p- Syl p (G) G p Syl p (G) 4.1.4 ( (2)). G (1) P Syl p (G) G p- Q g G Q g P (2) P, Q Syl p (G) P Q (3) Syl p (G) 1 (mod p). (1) G (Q, P )- 1.5.1 r r G = Qa i P = Q : Q a i P a 1 i P i=1 i=1 G = P G : P P Syl p (G) G : P 0 (mod p) r 0 G : P = Q : Q a i P a 1 i (mod p) i=1 Q p- Q : Q a i P a 1 i p- i Q : Q a i P a 1 i = 1 Q a i P a 1 i = a i P (2) (1) P, Q Q = g P Q = g P (3) N = N G (P ) (2) Syl p (G) = G : N P Syl p (N) N (2) P N G p- G (P, N)- l G = P b j N b 1 = 1 G : N N = G = j=1 l P b j N = j=1 l P : P b j Nb 1 j N j=1

4.1. 37 P N P : P N = 1 P : P b j Nb 1 j = 1 b 1 j P b j N b 1 j P b j = P b j N j = 1 j 1 P : P b j Nb 1 j 1 p- l G : N = 1 + P : P b j Nb 1 j j=2 Syl p (G) = G : N 1 (mod p) p- p- G p- G p- G p- p- 4.1.5. p, q p < q, p q 1 G = pq G ( 15, 35 ). P Syl p (G), Q Syl q (G) N = N G (P ) P N G N p pq N = p pq N = p Syl p (G) = G : N = q 1 (mod p) N = pq N = G P G Q G P, Q G p- q- 1 x G x o(x) o(x) G o(x) = 1 o(x) {p, q, pq} o(x) = pq x x = G G o(x) = pq x o(x) = p x = p p- p- P p- x P p 1 o(x) = q q 1 p 2, q 2 1 + (p 1) + (q 1) = p + q 1 < 2q pq = G o(x) = pq G 4.1.6. n 7 n S n 15. 15 15 7 1.8.14 S n (n 7) 15 (S 8 [5, 3] 15 ) 4.1.7. G p P Syl p (G) G H N G (P ) N G (H) = H

38 CHAPTER 4.. N G (H) H x N G (H) x H = H x P x H = H P, x P Syl P (H) P x P H h H x P = h P h 1 x N G (P ) H x hh = H N G (H) H 4.1.8. p = 2, 3, 5 5 S 5 p- p- 4.1.9. G p P Syl p (G) P G P G 4.1.10. p G H G P Syl p (H) P H P G 4.1.11. G H G p P Syl p (H) G = HN G (P ) 4.1.12. p 2p C 2p D 2p 4.1.13. p G p G p-

Chapter 5 5.1 G, H G H (g 1, h 1 )(g 2, h 2 ) = (g 1 g 2, h 1 h 2 ) (1 G, 1 H ) (g, h) 1 = (g 1, h 1 ) G H G H ( ) 5.1.1. G H = H G, G H K = (G H) K = G (H K) r G i, i=1 ( ) r r G i = G i i=1 G H G 1 = {(g, 1 H ) g G} G 1 = G G G H H G H 39 λ Λ i=1 G λ

40 CHAPTER 5. (1) G H G H (2) G H G H 5.1.2. G = r i=1 G i G g = (g 1,, g r ) (g i G i ) π i : G G i, π i (g) = g i G G i ι i : G i G, ι i (g i ) = (1,, 1, g i, 1,, 1) G i G π i ι i (g i ) = g i, (ι 1 π 1 (g))(ι 2 π 2 (g)) (ι r π r (g)) = g π i ι i = id Gi, r (ι i π i ) = id G i=1 5.1.3. G, H Z(G H) = {(a, b) a Z(G), b Z(H)} = Z(G) Z(H) 5.2 G H 1, H 2,, H r G G H 1, H 2,, H r ( ) (D1) i j H i H j (D2) G h 1 h 2 h r (h i H i ) G = H 1 H r 5.2.1. G H 1, H 2,, H r (E1) i H i G

5.2. 41 (E2) G = H 1 H r (E3) i H i (H 1 H i 1 H i+1 H r ) = 1. G H 1, H 2,, H r G g h 1 h 2 h r (h i H i ) f i H i gf i g 1 = h 1 h 2 h r f i h r 1 h 2 1 h 1 1 = h i f i h i 1 H i H i G (E1) G h 1 h 2 h r (h i H i ) (E2) x H i (H 1 H i 1 H i+1 H r ) x = h 1 h i 1 h i+1 h r x = 1 H1 1 Hi 1 x1 Hi+1 1 Hr = h 1 h i 1 1 Hi h i+1 h r (D2) x = 1 (E3) (E1), (E2), (E3) i j H j H 1 H i 1 H i+1 H r H i H j = 1 h i H i, h j H j h i h j h 1 i h 1 j = (h i h j h 1 i )h 1 j (h i H j h 1 i )h 1 j H j h 1 j H j h i h j h 1 i h 1 j H i h i h j h 1 i h 1 j H i H j = 1 h i h j h 1 i h 1 j = 1 h i h j = h j h i (D1) (E2) G g h 1 h 2 h r (h i H i ) h 1 h 2 h r = k 1 k 2 k r (h i, k i H i ) (D1) 1 = (h 1 h 2 h r )(k 1 k 2 k r ) 1 = (h 1 k 1 1 )(h 2 k 2 1 ) (h r k r 1 ) (E3) k 1 h 1 1 = (h 2 k 2 1 ) (h r k r 1 ) H 1 (H 2 H r ) = 1 h 1 = k 1 h i = k i i (D2) G H, K G = H K H ( K) G G 1 G (indecomposable) (decomposable) G H 1,, H r H i G H 1,, H r G 5.2.2. G = a a 6 = 1 H = a 2, K = a 3 G, H, K 6, 3, 2 H = {1, a 2, a 4 }, K = {1, a 3 } G = H K C 6 = C3 C 2

42 CHAPTER 5. 5.2.3. m, n C mn = Cm C n. C mn = a, C m = b, C n = c f : C mn C m C n f(a i ) = (b i, c i ) f(a i ) = (b i, c i ) = 1 i m n mn a i = 1 Kerf = 1 f C mn = mn = C m C n f f 5.2.4. p G n C p G (p-) p- GF (p) n 5.2.5. C 6 C 4 = C12 C 2 5.2.6. C 4 C 2 C 2 5.2.7. (g, h) G H g G h H (g, h) o(g) o(h) 5.2.8. D(G H) = D(G) D(H) G H G H

Chapter 6 6.1 6.1.1 (). C e1 C e2 C er e i+1 e i (i = 1, 2,, r 1), e i > 1 (i = 1, 2,, r) {e i } 12 6.1.1 C e1 C e2 C er r i=1 e i = 12 e i+1 e i {12}, {6, 2} 12 C 12, C 6 C 2 6.1.2. 16, 36 6.1.1 p- 43

44 CHAPTER 6. 6.1.3. p a p- G G H G = a H. G p- a G G H G = a H G G = 1 G 1 G = a H = 1 G a b a G/ a p- o(b a ) p- o(b a ) = p s c = b (ps 1) o(c a ) = p c a, c p a c p = a m p m o(a m ) = o(a) o(c) = p o(a) > o(a) o(a) p m m = m p d = ca m d a d p = 1 d d a = 1 π : G G/ d π a π( a ) = π(a) = a π(a) G/ d G/ d U G/ d = π(a) U U = π 1 (U) G = a U G a G, U G x a U π(x) π(a) U = 1 x a Kerπ = 1 a U = 1 g G π(g) = π(a) i u i u U π u U π(u ) = u π(a i u ) = π(a) i u = π(g) g a i u d g a i u d j j π(d j ) = 1 U u d j U G = a U G = a U 6.1.4. p p-. G p-a G 6.1.3 G = a H H H p- G 6.1.5.. p 1,, p l G P i Syl pi (G) G = l i=1 P i 6.1.4 P i G 6.1.6. G = l i=1 P i 6.1.7. G C e1 C e2 C er e i+1 e i (i = 1, 2,, r 1)

6.2. 45. 6.1.5 p 1,, p l G P i Syl pi (G) P i C fi,1 C fi,2 C fi,l(i) f i,j p i f i,1 f i,2 f i,l(i) f i,j+1 f i,j l(i) (i = 1, 2,, r) l l(i) < l i f i,l(i)+1 = = f i,l = 1 G j = r i=1 C f i,j G j r i=1 f i,j i f i,j+1 f i,j G j+1 G j G = G 1 G l 6.1.8. G 6.1.7 e 1,, e r. G = C e1 C e2 C er = Cf1 C f2 C fs e i+1 e i (i = 1, 2,, r 1), f j+1 f j (j = 1, 2,, s 1) m (G, m) = {g G g m = 1} a = (a 1,, a r ) C e1 C er a (G, m) i a i (C ei, m) (C n, m) = gcd(m, n) m S(m) m e i i T (m) m f j j e i+1 e i i < j m e j m e i f i p (G, p) (G, p) = p S(p) = p T (p) S(p) = T (p) (G, p 2 ) = p 2S(p2) + p S(p) = p 2T (p2) + p T (p) S(p 2 ) = T (p 2 ) l S(p l ) = T (p l ) e i+1 e i, f j+1 f j p l e i p l f i p e i = f i 6.1.1 6.2 ( ( ) )

46 CHAPTER 6. A B A B ( ) Z n Z/nZ 6.2.1 Z F = Z Z F 6.2.1.. F = Zx 1 Zx r 2F = {2f f F } 2F F F/2F F/2F = Zx 1 /2Zx 1 Zx r /2Zx r = Z/2Z Z/2Z F/2F = 2 r F/2F F r ( ) 6.2.2. f : A F A F B = Kerf B A. F = Zx 1 Zx r x i f(c i ) = x i c i A C = Zc 1 + + Zc r A = B C a A f(a) = r i=1 k ix i c = r i=1 k ic i f(c) = f(a) a = (a c) + c f(a c) = f(a) f(c) = 0 a c B c C a B + C A = B + C b B C c C c = r i=1 k ic i c B 0 = f(c) = r i=1 k ix i F k 1 = = k r = 0 c = 0 B C = 0 A = B C 6.2.3. F A A F ( ). F = Zx 1 Zx r F r r = 1 0 r > 1 a A a = r i=1 a ix i f : A Zx r f(a) = a r x r Imf = 0 A Zx 1 Zx r 1 A r 1 Imf 0 0 m Z Imf = mzx r

6.2. 47 f : A mzx r 6.2.2 A = Kerf C C C = A/Kerf = Imf = mzx r Z f Kerf Zx 1 Zx r 1 B r 1 A r 6.2.2 A T (A) A A A T (A) = 0 A ( ) 6.2.4. A A/T (A). a A a T (A/T (A)) 0 m Z ma = 0 ma T (A) 0 n Z 0 = n(ma) = (nm)a a T (A) a = 0 ( ) 6.2.5.. A a 1,, a r A = r i=1 Za i F = r i=1 Zx i x 1,, x r π : F A f( r i=1 k ix i ) = r i=1 k ia i B A π 1 (B) F 6.2.3 y 1,, y s π 1 (B) B π(y 1 ),, π(y s ) 6.2.6. A T (A) ( ). 6.2.5 T (A) a 1,, a r T (A) T (A) r i=1 k ia i (k i Z) a i n i (< ) 0 k i < n i ( ) 6.2.7.. A x 1,, x r A x 1,, x r y 1,, y m m i=1 Zy i = m i=1 Zy i x 1,, x m B = m i=1 Zx i A/B A/B x 1,, x r 1 i m x i B x i = 0 A/B x m+1,, x r j (m + a j r) x j Zx j B = 0 m i=1 Zx i + Zx j = m i=1 Zx i Zx j x j (m+a j r) x j (m+a j r) l A/B l 0 A A (l) = {a l a A} A (l) B B 6.2.3 A (l)

48 CHAPTER 6. f : A A (l) f(a) = a l f A A = A (l) A 6.2.3 6.2.8 ( ). A A = F T (A) F T (A) F T (A) ( ). A/T (A) 6.2.7 f : A A/T (A) Kerf = T (A) 6.2.2 A = T (A) B B B = A/T (A) B 6.2.6 T (A) F A T (A) A Z Z Z/e 1 Z Z/e r Z e i+1 e i A T (A) A 6.2.9. 18, 24, 30 6.2.10. C 4 C 6 C 10 C e1 C er e i+1 e i (i = 1, 2,, r 1) 6.2.11. p p 2 6.2.12. p p 3 6.2.13. A = Z Z/2Z (a, b) (a Z, b Z/2Z = {0, 1}) x = (1, 1), y = (0, 1) A = x y ( )

[1],, [2],, Akihide Hanaki (hanaki@math.shinshu-u.ac.jp) 2011/02/08 49

2, 34 G-, 29 G-, 27 n, 28 p-, 33 p-, 35, 7, 29, 7, 10, 14, 26, 30, 46, 25, 26, 7, 7, 7, 20, 16, 7, 16, 29, 42, 13, 7, 31, 33, 32, 16, 18, 7, 19, 7, 12, 26, 26, 7, 16, 15, 15, 15, 27, 30, 18, 24, 24, 9, 21, 28, 8, 40, 12, 48, 13, 15, 10, 19, 22, 12, 10 p-, 35, 36, 33 50

51, 10, 13, 9 G-, 30, 7, 14, 20, 20, 15, 7, 7, 17, 16, 20, 7, 15, 28, 28, 28, 9, 31, 32, 39, 40, 41, 46, 41, 41, 14, 14, 20, 30, 31, 20, 23, 47, 48, 47, 14, 25, 13, 40, 47, 7, 26, 9, 9, 16, 8, 10, 10, 7, 12, 7, 43, 7, 48, 46, 14, 14, 9, 46, 11, 11, 32, 24, 24, 15, 7