Белоногов. Задачник по теории групп

! % ) # %

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G1, . . . , Gn G = G1 × . . . × Gn G =

n

× Gi!

i=1

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. G = G1 ◦ . . . ◦ Gn

L . ! G G1, . . . , Gn

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G1, . . . , Gn #

(g1, . . . , gn)! gi G i {1, . . . , n}! $#H

(g1, . . . , gn)(g1, . . . , gn) = (g1g1, . . . , gngn),

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G = G1

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i {1, . . . , n}

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g → gˆ = (1, . . . , 1, g, 1, . . . , 1) (g Gi)

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A׈ B%

A B > ϕ >

A1 Z(A) B1 Z(B)% ? N := {(a, ϕ(a)−1) | a A1} A׈ B #06%@A$' A׈ B/N # "$

A B ϕ ϕ

A B # B A1 B1$

A ϕ B A ϕ B%

G = G1 × . . . × Gn. C πi : g = (g1, . . . , gn) → gi (g G) G Gi' Hπi

H G # $ H Gi

% .

% H G' Gi Gi'

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& %

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3 ( ) G
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( )	|ai| |ai+1|

# G o(am)$%

G ( )

( ) G = b1 × . . . × bn

% ? m = n |ai| = |bi| & i%

* G' ! ( )

( )D

nk(G) k

G%

0$ nk(G)

! & G:

Z2 × Z4 , Z2 × Z2 × Z4 , Z3 × Z4 , Zp × Zp2 × Zp3 Zpm × Zpn ,

p > m, n N%

@$ E ' & 0$'

& %

E & & & A B

# & 06%F$: 0$ nk(A) = nk(B) & k N'

@$ A B%

! G > k &

m (k, m N)% C & &

m G%

" - G = Zp ×Zp3 ' p > '

p2'

G%

# l, m, n N%

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@$ Zm × Zn Zk × Zd' k = [m, n] d = (m, n)%

$ #?$ . ' n

' n &

%

Z+ × . . . × Z+					Z+ × . . . × Z+					m = n%
	m					n

A' B' C >

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, Q+ & &

%

0$ 5 G = G1 × . . . × Gn >

G1, . . . , Gn				G	'	G G1		ˆ ˆ
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@$ 5				ˆ	ˆ		'	ˆ
								G = G1 × . . . × Gn Gi Gi &
	G = G1×				. . . ×Gn
i {1, . . . , n}		% #	ˆ
						Gi %$

G1, . . . , Gn > G%

0$ 5 G = G1 × . . . × Gn' & & i j Gi, Gj = Gi × Gj # ' ' gigj = gj gi & gi Gi gj Gj $%

@$	5 G = G1 . . . Gn Gi, Gj = Gi × Gj & & i
j' G = G1 × . . . × Gn D
		σ > {1, . . . , n}.
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G = Gσ(1) × . . . × Gσ(n) %
@$	ˆ	ˆ	ˆ ˆ
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! G = G1 × . . . × Gn% 0$ Z(G) = Z(G1) × . . . × Z(Gn)%

@$ G = G1 × . . . × Gn%

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Φ(G) = Φ(G1) × . . . × Φ(Gn)'

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:

$ G = (A × B) × C'

$ G = A × (B × C)'

$ G = A × B × C%

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@$ 5 G = A × B, A ≤ G, B ≤ G g = ab G a A, b B'

CG(g) = CA(a) × CB (b)%

$ 5 A r & '

B > s & ' A×B

r · s & %

G = A × B, A1 A2 > A' B1 B2 >

B%

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<$ 5 A1 A B1 B' A1 × B1 A × B A × B/A1 × B1 A/A1 × B/B1%

G = A × B' A B >

π(A) ∩ π(B) = ' H ≤ G% 0$ H = (A ∩ H) × (B ∩ H).

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<$ A B > & G% 6$ Aut(G) Aut(A) × Aut(B)%

F$ 5 A s1 & # & n1

& A$' B s2 &

# & n2 & B$' A×B s1s2 &

# & n1n2 & A × B$%

7$ G = A × B

A, B π(A) ∩ π(B) = ' 0$HF$ %

G = G1 × . . . × Gn%

0$ 5 ϕ > X G, !

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@$ 5 γi Gi Hi i {1, . . . , n}'

γ: (g1, . . . , gn) → (γ1(g1), . . . , γn(gn)) ((g1, . . . , gn) G)

G H = H1 × . . . × Hn

Ker (γ) = Ker (γ1) × . . . × Ker (γn)%

G = G1 × . . . × Gn%

0$ Aut (G) A' Aut (G1) × . . . × Aut (Gn)%

@$ 5 Gi & G & i {1, . . . , n}' Aut (G)

Aut (G1) × . . . × Aut (Gn)%

<$ 5 G1 = . . . = Gn' Aut(H) S '

S Sn S, A = A S%

G	= A × A' A > % A1 :=
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G G = A1 × Mϕ = A2 × Mϕ%

@$ * K G ' G = A1 ×K = A2 ×K

Mϕ ϕ Aut(A)%

G = A×B' A1 ≤ A' B1 ≤ B' |A1| = |B1| ϕ >

A1 B1%

Mϕ := {(a1, ϕ(a1)) | a1 A1}.

0$ Mϕ ≤ G ϕ > %

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<$ 5 Mϕ G' G/Mϕ &

' & A B' ' >

G G/Mϕ'

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' ˆ

' ˆ

∩

ˆ

ˆ

ˆ

ˆ

ˆ

∩

ˆ

G = A

B

A

A

B

B A

B = A1

= B1

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/A

B

A/A '

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Bˆ

B/B I

(a , 1) = (1, ϕ(a )−1)

1

∩ ˆ

1

1

1

&

A

ˆ

a1 A1% #J X := {(x, 1) | x X}

X

Y := {(1, y) | y Y }

Y B%$

! #K$

G = A × B, A1 ≤ Z(A), B1 ≤ Z(B), A1 B1 ϕ

> A1 B1 %

Nϕ := {(a1, ϕ(a−1 1) | a1 A1}.

? Nϕ G > G G/Nϕ'

<$ !

(a1, 1) = (1, ϕ(a1)) %

#) ' G ϕ

A B' : G = A ϕ B%$

" G1, . . . , Gn > G% 3

ˆ ˆ	#	ˆ	ˆ
		= G1	× . . . × Gn 06%06$
D := G1× . . . ×Gn

π := (D → G | (g1, . . . , gn) → g1 . . . gn).

ˆ

0$ π(Gi) = Gi%

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# ! :

#0$ π D G # ' ' G D/Ker (π)$I

#@$ !			γ		D			G	'	ˆ
ˆ										γ(Gi) = Gi
	&	i {1, . . . , n}				I
Ker (γ) ∩ Gi = 1

#<$ G = G1 . . . Gn%

$ 5 G = A B' A, B ≤ G' A ∩B Z(G) G A ϕ B'

ϕ > A ∩ B %

A' B G > % 3 :

	#0$	˜	˜	> #$
		G = A	B
˜	˜	˜		˜
A	B' A		A B BI

#@$ G A ϕ B > # "$ A B ϕ Z(A)

Z(B)

ˆ	#<$ ) ! γ A ×B G ' Ker(γ) ∩
	ˆ%
A	= 1 = Ker(γ) ∩ B

A B > ϕ >

A1 Z(A) B1 Z(B)% 0$ A ϕ B B ϕ−1 A.

@$ 5 A ' A ϕ B A B1 = B.

<$ 5 A B ψ >

A1 B1' A ψ B A ϕ B.

G = Zpk

ϕ Zpl , p > '

{

k, l

}

N, ϕ >

pn

& %

? G Zpk × Zpl−n , k ≥ l%

Gi = A ϕi B (i = 1, 2),

A = a p3 × x p2 ,

B = b p3 × y p2 ,

A1 = a1 p × x1 p ,

B1 = b1 p × y1 p

p2

, x1 = x

p

, b1

= b

p2

p

),

#p > ' a1 = a

, y1 = y

ϕ1 : a1 → b1, x1 → y1,

ϕ2 : a1 → y1, x1 → b1,

?

G1 Zp3 × Zp2 × Zp2 × Zp. G2 Zp3 × Zp3 × Zp × Zp.

A B > ' A1 ≤ Z(A)' B1 ≤ Z(B) A1 B1% ' A1

A # B1

B$% ?

Aϕ B A ψ (B)

& ϕ ψ A1 B1%

G1, . . . , Gn > ' Zi ≤ Z(Gi) Zi Z1 &

i {1, . . . , n}% ? ! H = H1 . . . Hn' Hi ≤ H'
n		βi		βi Gi		Hi #i {1, . . . , n}$ ' Hi Hj =
k							}
					{
	Hk = Zi		& i, j			1, . . . , n		i = j%
=1
	!	G = A B H = C D >

'

A C Z4m (m N), B D8, D Q8 |A ∩ B| = |C ∩ D| = 2.

? G H%

" G = A ×B, A ≤ G, B ≤ G H ≤ G% 5

HA # $ A'

H∩ A A%

# H > A B # % % H >

A × B' A B A

B

' $%

ˆ

| x X}

X A

%

X := {(x, 1)

ˆ

{(1, y) | y Y }

Y

B

Y :=

0$ ) ! # $ A0 A B0

B

'

ˆ

ˆ

H

ˆ ˆ

H ∩ A = A0

∩ B = B0 %

ˆ

ˆ

A × B.

@$ A0 A, B0 A A0 × B0

<$

ˆ

B

ˆ

A

%

H/A0

H/B0

6$

ˆ

ˆ

A/A0

B/B0 %

H/A0

× B0

F$ ) ! ψ A/A0 B/B0 '

H = {(a, b) A × B | (aA0)ψ = bB0}.

7$ |H| = |A||B0| = |A0||B|%

$ A B >

A/A0 B/B0% ? ψ A/A0 B/B0

	Hψ := {(a, b) A × B | (aA0)ψ = bB0}
		A		B	%			ˆ ˆ	, Hψ ∩
ˆ ˆ							Hψ ∩A = A0
	& 06%<9% * '					ˆ	ˆ	A/A0%
B = B0						Hψ/A0	× B0

#, Hψ ψ A B # B A/A1 B/B1$%$

G > ψ Zpk Zpl , p >

' {k, l} N, ψ >

pn & % ? G Zpk × Zpl−n , k ≥ l%

L A B &

&:

$ A = Z4, B = Z2I

$ A = Z4, B = Z6I

$ A B > & I

$ A = Z9, B = D6I

$ A = Z4, B = Z2 × Z2%

5 M N > G' G/M ∩ N G/M G/N %

G = ABC' A, B, C > G'

N = AB ∩ AC ∩ BC% ?

G/N = AN/N × BN/N × CN/N.

G = Gi | i I , Gi ≤ G I = {1, . . . , n}.

Ni = GG | j I \ {i} i I N = ∩ Ni% ?

j i I

n

G/N = × GiN/N.

i=1