Белоногов. Задачник по теории групп

>

A , !

" ! " ,

2 B " 9

C # C! C C!

C C ! # A

! :

3 " )

" C > C C > C !

D $ E !

" #

F

!

" "

! !

$" ! "

: # #

G ! ! ∩! ! \! " # #

.

E B "

|A| E A

{a1, . . . , an} E : a1, . . . , an

·{a1, . . . , an} E {a1, . . . , an} n ai = aj

i = j

{. . . | . . .} E . . . ! . . .

A := B E A B H A "

B

A × B := {(a, b) | a A, b B} E

A B

f : A → B E f ! f E ,$

A B! # (A, B, Γ)! A B E ! Γ A × B

" a A : b B #!

(a, b) Γ A E ! B E Γ E

, ,$

I f E ,$ A B! a A C A! f (a) af E a f !

f (C) := {f (c) | c C}! Cf := {cf | c C}!

f |C E f C

(A → B | a → . . . ) E ,$ A B! " a

A : . . .

a → . . . (a A) ,$" (A → B | a → . . .)! B

N E

Z! Q! R! C E '5! E $ $!

$!

< $ m n

(m, n) E # # m n! m|n ! m n!

π(n) E # n

A " ! !

ϕ ! ,$ N N ϕ(n) = |{m

{1, . . . , n} | (m, n) = 1}|

δij E ! ( i = j / i = j i j E "

B I a C!

|a| E a!

a¯ E ! 4 a

4 # $ :

G C C

3 k# m.n m.n(k)

) ! " , !

! " C3

# C $

G f ! # # x, y : M

# : M # xf y

M

J M # $# M !

H

( $ $! (a b) c = a (b c) " : a, b, c MD

. M # # !

: e #! e a = a " : a MD

0 , # : e M

" : a M M : !

: a! * : x #! x a = e

> ! K : M, )! M K

! K $ M # ( K 0 #

" A # # : # # : " 4 :

( (

J |M| M G = (M, )

G |G| I : !

! K

7 (M, ) ! "

: x y M ! x y = y x

M "

: G = (M, )

x G X G x M X M L ”(M, ) K C " H CM

$ C

"

# $H $" "

" · # " !

# # : "

" $, # (! : ! # : x!

" x " x−1 + "

H $" " "

+! # # : "

" $, # /! : ! # : x! "

x " −x

A # " ! !

# $

a K : < " $ n

n an : a : a0 = 1, ai+1 = aia i ≥ 0! an = (a−n)−1

n < 0 a :

: ! an = 1 n 2 n : #

: a o(a) I an = 1 "

n, a "

: o(a) = ∞ .

I G # : a! G

4 # : a

G = (M, ·) H = (N, )

ϕ M N !

ϕ(ab) = ϕ(a) ϕ(b) " a, b M.

=" M N " G H I , G H! G H "

G H I ϕ E , G H! ! !

ϕ $ G $ H ϕ(g−1) = ϕ(g)−1

g G G H

( 0/ ) ! ϕ

G H , !

$ G : g G :

7 (H, ·)! H E

X! · E H!

X I H

X! (H, ·) E ( .0 )

X SX !

X = {1, . . . , n}! n N! Sn f

n : x1, . . . , xn

: ( .6 , #

+ ! !

! (X, ·) !

$ ( .8! ( 0/

A : , "

H C+, R+, Q+, Z+! C·! R·! Q·! Zn # ( (0! ( (%! ( (8! ( (-! ! D(A) ( %5 ! GLn(C) ( %6 A ( 0.

$ " ! "

G H " G×H

J $# # $#

M ( &. K ( &%

N G = (M, ) E >

( G # # : e! 4 a e = a

a MD

. " a M M : x,

# a, 4 a x = e

N a, b, c E : I ac = bc ca = cb!a = b

A C # $C C C C # $C C C F H

M $# $# · #

! M : e #!

ae = a a M,

" a M : x M, #! ax = e

< M $# $# ·

H

( (M, ·) E D

.# a, b : M # ax = b

xa = b MD

0 # a, b : M # ax = bxa = b # # MD

M, ·! # . !

> ! K : $# $#

M E · E $ $ 4

; H ( M, · E D

. ϕ M M ! (aϕ(a))b = b = b(aϕ(a)) a, b M

· E $! 4

M N ϕ E

M N !

! ϕ , M, · (N, )

! G = (M, · E a M ) M "

" $" H