Белоногов. Задачник по теории групп

5 &

& & '

& & %

! G'

' G

Soc(G)% E '

G' ! Soc(G)'

& & G%

" ) ! G :

#0$ N G G

M ' G = N × M' #@$ G = Soc(G)'

#<$ G & &

& %

# ( G

& & ' &

G%

$ 5 G > gG = G & g G'

G = G%

#K$ * G # % %

& & & $ '

& & %

p > % L &

! & :

0$ A5 × Zp,

@$ A5 × Zp × Zp, <$ A5 × A5 × Zp.

G = G1 ×. . . ×Gn > Gi' ? N G N = N ' N = (N ∩ G1) × . . . × (N ∩ Gn)%

G = G1 × . . . × Gn > &

& Gi' & m &%

0$ 5 m ≤ 1'

G

Gi.

@$ E & & & & &

G m%

G > H G × G% 3 :

#0$ H ≤ G × G H GI

#@$ ! α β G Ker α ∩ Ker β = 1αβ = 1 ' H = {(α(g), β(g)) | g G}%

G > α, β, γ, δ > '

Ker α ∩ Ker β = 1' Ker γ ∩ Ker δ = 1% Gα,β = {(α(g), β(g)) | g G} Gγ,δ = {(γ(g), δ(g)) | g G} # !

> G × G' G$% 3 :

#0$ Gα,β = Gγ,δI

#@$ ! ϕ G ' α = ϕγ β = ϕδ%

! C ' A5' A5 × A5%

' & |Aut(A5)| + 2% C !

G%

	n
	i×
" G > Ω G =	=1 Gi ' Gi > Ω

# % % ! & & Ω $ Ω

G% S1 > & Gi'

Ω G1% ? Ω H G' Ω G1' S1%

# #? * H 3 H $% G > Ω'

! Ω

# % $%

G = G1 × . . . × Gm = H1 × . . . × Hn ,

Gi Hj > Ω

Ω G% ? m = n ! Ω

α G ' & !

Hj i {1, . . . , m}

G = G1 × . . . × Gi−1 × Hi × . . . × Hn Gαi = Hi .

#, ' Ω #

$ Ω'

Ω # ' $

%$

$ A B > G G = A × B%

α > A Z(B) C := {aaα | a A}% ? G = C ×B' 3 = A Ker (α) = A # % % α $%

#K$ G >

& ' G1, . . . , Gm, H1, . . . , Hn >

%

0$ G = G1 ×. . .×Gm = H1 ×. . .×Hn% ? m = n !

α G ' & !

Hi Gαi = Hi # ' ' Gi = Hi

0@%F=$%

@$ G = G1 × . . . × Gm% 3 :

#$ G &

&

I

#$ & & i j {1, . . . , m} !

& Gi Z(Gj ) # G

' (|Gi/Gi|, |Z(Gj )|) = 1$%

{Gi | i I} >

G% , ' G Gi i I

# : G = × Gi$' Gi G & i I

i I

G ' '

' & &

Gi%

0$ 5 |I| = n N {Gi | i I} = {H1, . . . , Hn}, G = × Gi

i I

G = H1 × . . . × Hn%

@$ 3 :

#$ G = i×I Gi'
#$ G = Gi | i I J I
Gi | i J = i×I Gi .
	Gi' Z(G) = i×I	Z(Gi) G = i×I Gi%
<$ 5 G = i×I

I > i I

Gi # Gi = Gj i = j$% C
M > & f I i I Gi &'
$ f (i) Gi & i I
$ f (i) = 1 " i I%
C M ·
	(f · g)(i) = f (i)g(i) (f, g M).
0$ (M, ·) > % # i×I Gi
# "$	Gi i I$%
@$ 5 I = {1, . . . , n}' i×I Gi				G1 × . . . × Gn% (
= D						˜
<$ ) !							| i I}		G = i×I Gi
					{Gi
' ˜	˜		&	i I		'	G = i×I	˜
Gi G, Gi Gi								Gi%
6$ 5 G =	i×I Gi ' s SI s˜ : f →
s−1f (f G) G # s−1f (i) = f (s−1(i))$%
{Gi | i I} {Hj | j J} > %
0$ 5 I = J Gi Hi			& i I' i×I Gi j×J Hj %
@$ 5 I ∩ J = ' (i×I Gi) × (j×J Hj ) i ×I J Gi' Gi									= Hi

i J%

G = (M, +), M > & &

(a1, a2, . . .) & ' + > M,

(a1, a2, . . . ) + (b1, b2, . . . ) = (a1 + b1, a2 + b2, . . . ).

0$ G > %

@$ G × Z+ G.

<$ G × G G.

#) ' A × B A × C ' B C%$

I Gi > 06%7< F > &

f I Gi &' f (i) Gi & i I% C

i I

F · (f · g)(i) = f (i)g(i) (f, g M). ?
(F, ·) > ' i×I Gi > %
#,	(F, ·)	¯	Gi
		i×I

Gi i I%$

$ % % %

3 " '%! G = A B !

G A B! G =

AB! A G A ∩ B = 1

A B E ϕ E , B

Aut(A) > A × B $#

ϕ(b−1)

(a1, b1)(a2, b2) = (a1a2 1 , b1b2)

# (& ( )

A ϕ B A B #

˜

ϕ$ a˜ := (a, 1) a A b := (1, b) b B

→ → ˜

) iA : a a˜ (a A) iB : b b (b B) "

, (& . " A

B G

G E H = G id Aut(G)! id E #

, Aut(G) Aut(G) 7 ! " H #

: (g, 1) : g (g G) : (1, a) : a (a A)!

Hol(G) G

A! B E ! X E P E , B

" SX H := × A

x X

|X| : AD : "

,$ h X A ! h(x) = 1 :

x X 3 (% 50 , σ SX Aut(H)

#! σ(s) : h → s−1h (s−1h(x) = h(s−1(x)) ) A !

X = {1, . . . , n}! : H "

# h = (a1, . . . , an) ai A σ(s) : (a1, . . . , an) → (as−1(1), . . . , as−1(n)) > $ Pσ , B Aut(H)!

A P B := H Pσ B A !

B P ! ! P A B I

: H := ׯ A!

x X

A¯PB := H Pσ B P A B A

H

A A P B

A B! B ≤ SX P E B SX !

A r B! P E B !

r " ! B

"

A B E G H = AG 7 ! G

#$ A B

G = A B! G = H B H
G! 4 A G H = K× K! K =
{A}G	K

2 '8 ! G # 9 !

G = N H! N G! H ≤ G H ∩ Hg = 1 g G \ HD N

H E G

) B A ϕ B! 4 # ! #

#

G = A ϕ B E A B

˜

a˜ = (a, 1) a A b = (1, b) b B

˜

"

( ) iA : a → a˜

(a A) iB : b →

b (b B)

,

A

B

˜

˜

G

A

B

˜

= a ( )

. G = A˜ B˜

a A b B

a˜

b

ϕ b

A B E ; H

( G = A1 B1! A1 A B1 BD

. G A ϕ B

, ϕ

G = A B

( bG ∩ B = bB

" b B

. =" : B! 4 G! B

0 CB (a) = B ∩ Ba " a A

% L ( K 0 ! ! ! G =

AB! A G! A ∩ B = 1

G = A B! b B B1 B

( CG(b) = CA(b) CB (b)

. CG(B1) = CA(B1) CB (B1)

0 NG(B1) = NA(B1) NB (B1)

G E 9 N

H

( CG(h) H h H \ {1}.

. N = (G \ HG) {1}

0 CG(n) N n N \ {1}.

! G = N H ; H ( G E 9 N !

. # : N

: H

" G = A B! A E

G BG = 1
( A E	G
. CZ(B)(a) = 1	a A \ {1}

0 I Z(B) = 1! A Z(B) E 9 A

# ) " ,

Z+ ϕ Z+

$ ) , " H

( Z2!

. Z3!

0 Z2 × Z2! % Z+

Hol(S3) S3 × S3

I G E Aut(G) = Inn(G) Z(G) = 1! Hol(G) G × G

3 " G H

( G E D

. # ! G #

X! X = G × CX (G)

H = {(a, g) | g G, a Aut(G)} $#

(a1, g1)(a2, g2) = (a1a2, g1a2 g2).

> H Hol(G)

H := Hol(G) = G A! A = Aut(G) ( CH (G) = {gi−g 1 | g G} G

. G Inn(G) = GCH (G) = CH (G) Inn(G) H = CH (G) Inn(G) 0 Z(H) = CH (A) = {g Z(G) | ga = g}

H := Hol(G) = G A! A = Aut(G) ( NH (A) = Z(H) × A

. CH (A) = Z(H) × Z(A)

0 NH (A)/CH (A) Inn(A)

! I G = A B CB (A) = 1! G ,

Hol(A)! # A

" G E ! Z(G) = 1 Aut(G) = I K!

I = Inn(G) ; , GH Hol(G) = G (I K) >

G K I K

# G = A B! 4 B # A CB (A) = 1 A B

> B B! G E 9 A

$ G H E ! G = A B! α E , A → Hβ E , B → H ! α(ab) = α(a)

a A b B >

γ : ab → α(a)β(b) (a A, b B)

, G → H [α(A), β(B)] = 1

A B E ! A1 ≤ Z(A)! B1 ≤ Z(B) ϕ E

, A1 B1 ! , γ : A → Aut (B) #! γ(A0) = 1 γ(A) # B1

( A G = B γ A

N= {(a1, ϕ(a−1 1)) | a1 A1}

. < " X G X X

, G G/N >

G = BA, B B, A A, B G, B ∩ A ≤ Z(G), A ∩ B A1 B1 N.

+ " $ !

G ! (ϕ, γ)

B A

$ ! ,

Q8

G = A B! B ≤ Sn

n N > "

,

b →

˜

!

A1, . . . , Ann

G

˜

b

B

G

˜!

i×

!

b

G = H B

Ai

Ai = Ab(i)

i

H =

=1

G = A P B! P E , B SX ! H E G ! G = H Pσ B! σ E , SX H! 4 # iH iB E

H

B

G

˜

˜

˜

˜

iH (h) = h = (h, 1)! iB (b) = b = (1, b)!

H := iH (H)

<

"

x X

B := iB (B)

˜

˜

| h(y) = 1

y X \ {x}}.

Ax := {h

H

(

Ax

˜

Ax

A

x X

≤ H

. ˜

H = x×X Ax

˜

0

˜

˜

b

= AP(b)(x)

x X

G = H B

Ax

G, A, B E n N ; H