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Белоногов. Задачник по теории групп

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# # M E ! H E G ;

H

( {Mg | g G} = {Mh | h H}! . G = NG(M)H

#$ G = ANG(M)! A ≤ G M G > G

" N "!

M N A, M .

# C = AB! A < G B < G!

" #H

( A " " B!

. A ∩ B " ! " B! 0 CA(B) = 1

> G

# G = AB , A B E G, A0 A ≤ G .

! |G : B| |A0| >

AG0 B.

# M E ! A B E G

3 " H

( {Ma | a A} = {Mb | b B}!

. A, B = AN = BN ! N = N A,B (M)

# ) < . ./ " H

I

A

 

B

E

G

 

˜

 

 

 

 

 

 

A

G! A!

 

 

 

 

 

 

 

 

 

˜

˜

 

 

 

 

 

 

 

AB ∩ A = A(B ∩ A)

 

 

 

 

 

 

 

˜

˜

∩ B)A.

 

 

 

 

 

 

 

A

∩ BA = (A

 

 

 

# A A4 # A B !

A4 = ABA

˜

#

G = ABA! A B E G! A ≤

≤ G

> ˜

˜

 

A

A = A(B ∩ A)A

 

# ! A, B, C E # ! AC =

BC

( AD = BD ≤ G, D = N A,B (C)! ! N A,B (C) = 1,

A = B

. 3 C1 C ! AC1 = BC1 ≤ G

# " A0 A ≤ G B0 B ≤ G

( A0B0 ∩ A ∩ B = (A0 ∩ B)(B0 ∩ A)

. I A0 B0! A0 ∩ B B0 ∩ A

# #

G = AB, A ≤ G, B ≤ G, A0

A, B0 B I

y

y

A0xB0 = B0 A0x x, y G, A0B0 = B0A0.

#$ A, B, H E G G = AB. 2

H = (A ∩ H)(B ∩ H) H G

 

# A, B H E G! G = AB = (A∩B)H.

>

 

H = (A ∩ H)(B ∩ H).

 

 

 

 

# G = AB, A ≤ G B ≤ G I

 

A B A ∩ B !

 

 

 

NG(A ∩ B) = NA(A ∩ B)NB (A ∩ B).

 

# G = AB, A, B E G, A ∩ B = 1

H ≤ NB (A). >

 

 

 

NG(H) = NA(H)NB(H).

 

# G = AB, A ≤ G B ≤ G, N

G ;

 

H

 

 

( AN ∩ BN = (A ∩ B)N ;

 

. N = (A ∩ N )(B ∩ N ).

# G = AB, A B E

G " : A " : B > " # # N G

N = (A ∩ N )(B ∩ N ).

! : !

#

# ) 8 %& " G = AB,

A B E G N G. I A/A ∩ N B/B ∩ N E

" : #

" : #!

N = (A ∩ N )(B ∩ N ).

# ! G = AB, A ≤ G B ≤ G, G0 = A0, B0 !

A0 A B0 B

( I A B G0 !

NG(G0) = NA(G0)NB (G0).

. I A B " : A " : B!

G0 = (A ∩ G0)(B ∩ G0).

# " A, B, H E G

S4! AB = BA ! (A ∩ H)(B ∩ H) = (B ∩ H)(A ∩ H) G = AB

# # G E ! " "

H

G H "! H ∩ Hx = 1 x G \ H

! NG(H) = H)D

G N "! G = HN H ∩ N = 1

> N G ! ! G E 9

N H F ! N E

G #

#$ G = AB E ! A B E

G, A <· G! A 4

# ! B < G, A ∩ B = 1 AG = 1. > G E 9

B A

# n N : D2n #

9 O

# # 9

# G = AB ≥ AB1! A ≤ G B1 ≤ B ≤ G ( I B1 <· B! AB1 <· G

. I AB1 <· G! B1 <· B O

# A ≤ G H = Ag1 , . . . , Agn ! g1, . . . , gn E : G B E # : B

G !

( AB = G

. AtB = BAt t n giH

> NG(B ) H

i=1

# A ≤ H ≤ G B E B G !

( AB = G

. AhB = BAh h H > NG(B ) AH

# = A B E G !

AB = G AgB = BAg g G > G "

! " A B

# ! A ≤ B ≤ G! G E AgB = BAg

g G > " N1, . . . , Nk (k N) G !

N1 . . . Nk G A ≤ N1 ≤ B

# " I p G

G! G $ p2.

# # A E G I

A G! A

G J 7

# $ G = a b ! o(a) = o(b) = 4 a2 = b2 > ( ab {a, a1};

. G = a × ab Z4 × Z2! G Q8

1

A " : ,

G E !

p E

p E ! p

I pa E p! |G| (a N {0})!

" pa G

p G J p

G Sylp(G)

) G E : p G

p! |G| > ! CP E p GC CP E p

G p! |G|C J G

Syl(G) p G

p G!

p G Op(G) " " p G π(G) # G 3

3 (/ (! (/ . π(G)

# : G A ! G p

G p 7 G

! (G)| ≤ 1 ! (G)| = 2

π E

π E : !

π 7

π ! 4 π: #

# {p} p

p ! " p

" (/ 0 H # G

& G! |H| |G : H| !

π&! E π |G : H| π

G ! Cπ C C πC "

#

$ > 3 G E p E

( G p P E

. p G # !

4 # P.

0 A p G G

|Sylp(G)| = |G : NG(P )| ≡ 1 (mod p).

$ π E 7 G π

# |G| π

$ H E G

( H G H E |H| G

A H G p G " p"

. I H K G! H G 0 I H G K ≤ G!

|H| |K| H ≤ K!

|K| |H| ! K ≤ H

$ P E p G

( I P < S Sylp(G)! P < NS (P )

. P Sylp(G) P Sylp(NG(P )).

$ P Sylp(G) N G > ( N ∩ P Sylp(N )!

. P N/N Sylp(G/N )!

0 NG(P )N/N = NG/N (P N/N )

$ H E G

( + P Sylp(G) ! H ∩ P Sylp(H)

. J G

H ?

0 I p P H

K! 4 # H! H K : NG(P )

$! I P Sylp(G) NG(P ) ≤ H ≤ G! ( NG(H) = H!

. |G : H| ≡ 1 (mod p)

$" N G P1 Sylp(N )

( 9 G = N · NG(P1).

. I NG(P1) ≤ H ≤ G! |G : H| ≡ 1 (mod p) 3 ! H

" p" G

$# S E p G r E

p G! S > r ≡ 1 (mod p)

$ $ k E : p #

G > k ≡ −1 (mod p)

$ P Sylp(G) n = |G : P | I # (n n ( " p! P G!

P G

$ 3 : -

(56O

$ p q E

( pq pq2 " "

. I |G| = pq2 p | q − 1! G pq. 0 I p | q − 1! pq. % 3 pqO

$ =" 22 · 52! 23 · 52! 23 · 7! 22 · 72! 33 · 5! 54 · 7! 7 · 11 · 13! 22 · 7 · 23 " "

$ G E paqb! p q E

a, b N I # qb

( " p! G " pa

pa−1qb

$ |G| = pan! (p, n) = 1 (a, n N)! p E W W

a, b N !

# n ( "

p!

G H n

> G p! H Op(G)H G

$! ( 7 10000

. 2 · 52 · 7 3 · 7 · 19 "

" -

$" 2 0/! 06/! %8&! &./! &%5 L

A ! ! " 0/

0! & (&

$# ) ! "

"

:

 

$ $ I # G "

m G

xm = 1 m #! G $

$ G E ! # "

A B A ∩B = 1 AG ∩ BG = 1 > G

! 4 :

Z(G)

$ ! " 4

: ! !

3 Z

$ Sylp(G) = {P1, . . . , Pn} |Pi : Pi ∩ Pj | ≥ pd i = j > |Sylp(G)| ≡ 1 (mod pd)

$ G E pαn, p E ! (p, n) = 1!n ≡ 1(pk) k! 1 ≤ k ≤ α > G

" p" ! ! #

 

 

pα−k+1 ! p

$ ) (/ .0 " D

n = |NG(H)|! r E

H E h G,

# H 4

# ! 4

 

 

 

 

(

G

 

 

 

n mod

h2

D

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

|

 

| ≡

 

 

 

 

 

r

2

 

 

 

 

 

 

.

|{

H

}

G

| ≡

1

mod

 

h

 

 

 

 

 

 

 

2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(h , nr)

 

 

 

G

 

 

 

 

 

 

H E

 

 

 

 

 

 

 

 

0 I

 

 

 

 

 

!

 

 

 

 

 

 

 

 

 

 

 

 

 

 

|{H}G| ≡ 1

mod

h

.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r

 

$

H ≤ K ≤ G! G E ! H E

K |G : K| < p! p E π(H) > H G

$! G E .% >

G E 0 !

G E . G/Z(G) A4!

G S4

$" ) S4

$# 7 S4 0 H

( A4 !

. . ! , D8 !

0 % ! , S3

$ $ Z p Sp (p − 2)! 3

! (p − 2)! 1(mod p)

 

i=1[ pi ]

$ P Sylp(Sn) > |P | = p

! a =

a

 

n

n