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p-sylow :15114075 30 2 20

1 2 1.1................................... 2 1.2.................................. 2 1.3.................................. 3 2 3 2.1................................... 3 2.2................................ 5 2.3...................... 6 2.4......................... 7 2.5................................... 8 3 T n p-sylow 9 3.1 p-sylow................................. 9 3.2 T 4.................................... 10 3.2.1 3-Sylow............................. 10 3.2.2 2-Sylow............................. 11 3.3 T 6, T 8................................... 12 3.3.1 2-Sylow............................. 13 4 15 4.1................................ 15 4.2................................... 15 1

1 1.1. [3].. (p-sylow ). 1.2.... 5 4 6 8 12 20. T n (n = 4, 6, 8, 12, 20) G n. p n p p m. G p m G p-sylow.. T n p-sylow 1. T 4 p-sylow p = 2 : 2-Sylow : ( 4 ). 2-Sylow : ( 4 ). 2-Sylow 3. P = 3 : 3-Sylow : 6 ( 1, 2 ). 3-Sylow : 6 ( 3 ). 2. T 6, T 8 p-sylow p = 2 : 2-Sylow : ( 6 ). 2

2-Sylow : ( 6 ). 2-Sylow 3. 2-Sylow T 4 3-Sylow 6 3-Sylow 6. T n p-sylow. p-sylow,. 1.3 2.. 3 p-sylow. p-sylow. T 4 T 6, T 8 p-sylow. 4. 2 2.1. 1 ( ). G (G ). ( ) G a, b, c a (b c) = (a b) c (1). ( ) G e G a. e G. a e = e a (2) 3

( ) G a G b a b = b a = e (3). G a 1 b = a 1. a b ab. 2 ( ).. G G. 3 ( ).. G. G = g = {g n n Z} (4) 4 ( ). X X X. X S(X). S(X). X n = {1, 2,, n} S(X) n S(X) S n. S n := {f : X n X n f } (5). S n n A n. A n := {f S n f } (6). f ( ) 1 2 n f = f(1) f(2) f(n) (7). 5 (3 S 3 ). S 3 6. ( ) ( ) = e = (12) 2 1 3 ( ) ( ) = (23) = (13) 1 3 2 3 2 1 ( ) = (123) 2 3 1 ( ) = (132) 3 1 2 (12) (123) 3. 6 (3 A 3 ). A 3 3 ( ) ( ) = e = (12) 2 3 1 ( ) = (132) 3 2 1 4

7 ( ). G H G. G x H h G. xh := {xh h H} (8). xh x. Hx. 8. G H G/H (G H) H G. (G H) = G/H (9).. G/H = G H (10) 9 (Lagrange ). G H G. H H G G. [2 (11.1) ]. 10 ( ). 2 G G φ : G G (11) φ(ab) = φ(a)φ(b) (a, b G) (12) φ G G. φ ( ) φ ( ) φ φ. 2 G G G G. G = G (13) 2.2. 11 ( )... 5

.. 5 4 6 8 12 20. T n (n = 4, 6, 8, 12, 20) 12 ( ). n E n E n f. f E n x, y x y f(x) f(y). d(x, y) = (f(x) f(y)) (x, y E n ) (14). 13 ( ). T n (n = 4, 6, 8, 12, 20) T n T n Q(n). T n P (n). 2. 2.3. 14. O 1. O 2. O 3. O 3.. (. 3 1, 2, 3.) ( ).. O O. O. O 2 ( ). 2 1 a. a. a O. 6

a b b. a. a b b b (b ). b ( ) ( ). O. a O. 15. O 1. O 2. O 2. ( ).. O O. O. O...... ( ) ( ).. (1) (2). (1) O. (2). O. O. 2 O. 14 1 2. 2.4. 16. P (4) = A 4 (4 ) ( ). 4 1, 2, 3, 4. 4 (P (n) ). 4 1, 2, 3, 4. 13 3.. 1. O (O ) 2 3 π, 4 3 π 7

2. O π 4 O O. 1 4. 8. 2 3. 3. e P (n) 12.. S 4. P (n) ( ) A 4 ( ). A 4. P (4) = A 4. 17. Q(4) = S 4 ( ). 4 1, 2, 3, 4. 4 (Q(n) ). 4 1, 2, 3, 4. Q(4) ( ) S 4 ( ) 2. Q(n) 1. Q(n). Q(4) = S 4. P (n), Q(n) [2, p. 50,63]. n 4 6 8 12 20 P (n) A 4 S 4 S 4 A 5 A 5 Q(n) S 4 S 4 Z 2 S 4 Z 2 S 5 S 5 : P (n),q(n) 2.5.. 18 ( ).... 4 4 ( ) 6 8 8

12 20.. 19.. { P (6) = P (8) Q(6) = Q(8) { P (12) = P (20) Q(12) = Q(20) (15)... 3 T n p-sylow T n p-sylow p-sylow. p-sylow. T 4 T 6, T 8. 3.1 p-sylow 20 (p-sylow ). G H G. p p m G G H H = p m H p-sylow. 21 ( ). G H G. H. N G (H) = {g G ghg 1 = H} (16) 22 (Sylow ([2] (20.3) )). n p G p-sylow. n p = G : N G (H) = G (17) N G (H) n p 1 (mod p) (18) p-sylow [5]. 9

23 ( ). x X x G x (stabilizer) Stab G (x).. Stab G (x) = {g G gx = x} (19) 3.2 T 4 T 4 1. Q(4) = S 4 G. G = 2 3 3 G 2-Sylow 3-Sylow. T 4 1, 2, 3, 4. Q(4) S 4. 3.2.1 3-Sylow x, y 0 < x, 0 < y, 0 < x + y < 1 (20). U 3 3-Sylow. U 3 = 3. 3. O 3. U 3 A(x y ). A T 4 6. A Stab G (A) = {e, (123), (132)} = U 3 (21) 1: 2: 3-Sylow U 3 N G (U 3 ) = {e, (12), (13), (23), (123), (132)} (22) 10

. (12), (13), (23) 3 4 O 2 4 O 1 4 O. B(A x, y x = y ). B T 4 6. 3: A B. x y 6 x = y 6. U 3 6 N G (U 3 ) 6. 3-Sylow n 3 N G (U 3 ) = 6 n 3 = G N G (U 3 ) = 4 (23) B 1 4. 3.2.2 2-Sylow U 2 G 2-Sylow., U 2 = 8. 4 C. C T 4 1 2 2 3 3 4 4 1 11

. C Stab G (C) = {e, (13), (24), (13)(24), (12)(34), (23)(14), (1234), (1432)} = U 2 (24). 4: 2-Sylow U 2 N G (U 2 ) = {e, (13), (24), (13)(24), (12)(34), (23)(14), (1234), (1432)} (25) N G (U 2 ) = U 2 N G (U 2 ) U 2 C. n 2 N G (U 2 ) =8 n 2 = T 4 C 3. G N G (U 2 ) = 3 (26) 3.3 T 6, T 8 U 2 A T 4. G=Q(6) Q(8) S 4 Z 2. G = 48 = 2 4 3 2-Sylow 3-Sylow. 6 1, 2, 3, 4. Q(n) S 4. Q(n) +1 1 Z 2 = {±1}. 12

5: 3.3.1 2-Sylow U 2 G 2-Sylow., U 2 = 16. 6 A. A 4 2 5., A Stab G (A) ={(e, ±1), ((13), ±1), ((24), ±1), ((13)(24), ±1), ((12)(34), ±1)((14)(23), ±1), ((1234), ±1)((1432), ±1)}( ) 6: 2-Sylow U 2 Stab G (A) ={(e, ±1), ((13), ±1), ((24), ±1), ((13)(24), ±1), (27) ((12)(34), ±1), ((14)(23), ±1), ((1234), ±1)((1432), ±1)} (28) N G (U 2 )( ) (29) 13

N G (U 2 ) = U 2 N G (U 2 ) U 2 A. n 2 N G (U 2 ) = U 2 = 16 G n 2 = N G (U 2 ) = 3 (30) A 3. 14

4 4.1 Geogebra.. p-sylow.. T 4 p-sylow, (p=2,3 ) T 6, T 8 2-Sylow, Sylow (T 4 p=2 ) T 4 2-Sylow 4.2 T 6, T 8 3-Sylow,. T 12, T 20 p-sylow,. p-sylow,.. 1. [1] 30 1989. [2] M.A 2007. [3] 2015. [4] 1987. 15

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