Белоногов. Задачник по теории групп

$ G		|	= pαn! (p, n) = 1!		{	α, n	}	N n > 1 I
|			n
Op(G) = 1! α <				! p = 2 α < n − 1
	p − 1

$ 3 "

H

D6 × Z2!

D6 × D6 O

$ P E G = GLn(Zp) (n N)!

4 $ '5 > P Sylp(G)

$ + # !

3 ! " I G E ! H ≤ G P Sylp(G)! : g G #! H ∩ P g Sylp(H)

$ < #

! (/ 0% (/ 0&

$! G = AB! A ≤ G B ≤ G

( 3 " p P A p

Q B ! P Q Sylp(G)

. I A1 E p A! B1 E p

B! A1, B1 E p

$" I G = A × B! A B E G! P Sylp(G)!AP ∩ BP = P

$# G1 G2 E ! G1 n1 p

! G2 n2 p 3 p

G1 × G2O

$ $ G E ! M G P Sylp(M) I P G!

M! P Sylp(G) Op(G) Sylp(MG). 2 ! MG = Mg

g G

$ P Sylp(G) > " : CG(P )!

4 G! NG(P )

$ P Sylp(G) x E p: G I x

Z(P )! x $ Z(P )

$ P Q E p G I A ≤ Z(P ) A Q! A ≤ Z(Q)

$ a E p: G |G : CG(a)| = pα! α N {0} > a Op(G)

$ ( P Sylp(G)! P1 P2 E

P I P1 P2 G! P/P1 P/P2.

. L ( ! 4

P1 P2 4 H P1 P2
G k E			p		k
$ P		Sylp(G)! k E 4 :
P k	!			: >		≤
| |

$! I " p G

(! G " p: p

$" P Q E p G

! P ∩Q	p G
P ∩Q	#

p G

( Np(P ∩ Q) NQ(P ∩ Q) E p

NG(P ∩ Q) ! NG(P ∩ Q) p# #

. CP (P ∩ Q) CQ(P ∩ Q) E p CG(P ∩ Q) 0 P Q : NG(P ∩ Q)

$# N E p G >

" p G/N DN/N !

D E p G

$ $ ( N E

G! p > N p

. Φ(G) E G > Φ(G) #

$ H E G P E

p! !

4 H > NG(P ) # !

4 H ] ^Q_`*abcd

$ G = HK! H K E G! |H| ≤ |K|

K G > x G \ NG(K) P K ∩ Kx ! NG(P ) ! 4

H! ! 4 K

$ G! 4

! !

( H K E : ! G

!

. G = P Q! P Q E

Q $ ] ^Q_`*abcd

$ I N E p G P E

p G!

( NG/N (P N/N ) = NG(P )N/N !

. CG/N (P N/N ) = CG(P )N/N

$ N E p G

( Z 4 p: G/N

4 p: G

. I x E p: G! CG/N (xN ) = CG(x)N/N !

! |CG/N (xN )| = |CG(x)|/|CN (x)|						α|Sylp(NG(A))|
$	A P Sylp(G)	>	|{A}	NG(P )	| =		! α
						β
E ! 4 A						P ! β E

p G! A

$! P Sylp(G) x P >

|CG(x)|p = max |CP (t)|.

t P ∩xG

$

$" P Sylp(G)! z Z(P )

|zG ∩ P ∩ P g | < p g G \ NG(P ).

> z $ # # p G!

#

$# X # G p

G! |X| |G : NG(X)| E p X p !

# # p# G

I A B p!

A × B

$ $ H ≤ G I H # #

# G! NG(H) " "

G! " " H! H G +

$ I H G #

# # G! H

# # G! H G +

$ H < G! H # #

# G > NG(H) > H !

|G : H|! |NG(H) : H|

G E Sn! P Sylp(G) P1 Sylp(H)! H E # G > NG(P ) # F ix(P ) NG(P1) #

F ix(P1)

2 3

7 , 4

! , E 4

J : , , # G

" End(G) Aut(G) ! Aut(G)

# # SG " g G

i(g) : x → g−1xg (x G)

, G! # 4

! " : g i iG

A , G

" " Inn(G) Aut(G)! ,

G/Z(G) (( (5 &

G 9 Out(G) := Aut(G)/Inn(G)

" & G

7 G "! Aut(G) = Inn(G) Z(G) =

1

, ϕ G !

# : G

H E G ϕ End(G) 7 ! H ϕ

! ϕ ! ϕ!

Hϕ ≤ H I Hϕ = H! ! ! ϕ |H End(H) : , H!

ϕD H G! ϕ

4 # : , G/H

(( .. H A ! A !

A ≤ Aut(A)! α " α A I H

, G! H

& G 7 ! "

! &

, α G !

" , > ! CAut(G)(Inn(G))

, G

I A E ! End(A) $

$#H

ϕ + ψ : a → aϕ · aψ, ϕ · ψ : a → (aϕ)ψ

(ϕ, ψ End(A), a A) (( &

" "

K Aut(K)

Ω E ! G E ϕ E

Ω End (G) > # (G, Ω , ϕ)

Ω 7 ! G Ω ! ! 4

! ϕ !

gω gϕ(ω) g G ω Ω G E Ω

" ϕ I H ≤ G Hω H Hϕ(ω) H

ω Ω! ! H Ω G ! H

Ω G I N E Ω G!, G/N Ω# "

ϕˆ : ω → (gN → gωN ) (ω Ω, g G) ) α Ω GΩ H Ω !

gω α = gα ω g G ω Ω.

HomΩ(G, H) E Ω, G H ; Ω "

! Ω ϕ E , Ω Aut (G)

A ! A ≤ Aut (G), X G N M ≤ G "

H

CA(X) := {α A | xα = x x X};

CA(M/N ) := {α A | (mN )α = mN m M}; NA(X) := {α A | Xα = X};

CG(A) := {g G | gα = g α A}.

I CA(Y ) = A! Y X M/N ! ! A

Y

" # ϕ C· "

: , ! ! , O 2 # Ker(ϕ) (C·)ϕ

( ϕ : z → zn! n E 2 #

U Hn C·! 4 % &- % &6
. ϕ : z →	zn		! n N
	|z|n
0 ϕ : z = a + bi →				z	= a − bi (a b E #
% ϕ : z → cz!			c E , # : C·
& ϕ : z → |z|
5 ϕ : z → z−1

" # ϕ C+ ": , ! ! , O 2 # Ker(ϕ)

(C+)ϕ
( ϕ : z = a + bi →			= a − bi (a, b R)
		z
. ϕ : z → Rez	z)
0 ϕ : z → zn!	n E $
% ϕ : z → nz!	n E $
& ϕx : z → zx!	x E , # : C+

G E ! α Aut(G) X G

( CG(X)α = CG(Xα)

. NG(X)α = NG(Xα)

0 G! 4 X G, α

G! 4 Xα G

I N G α Aut(G)! G/N G/N α

< # A End(A) $

$#H

ϕ + ψ : a → aϕ · aψ ϕ · ψ : a → (aϕ)ψ (ϕ, ψ End(A), a A).

: Aut(A) = End(A)·

K E $

( < x K αx K K

αx(y) = yx > {αx | x K} $ End(K+)! ,

K

pn−1

. I K E $ $#! Aut(K+) !

, " K·

! 3 " , $ H

End(Z+) Z Aut(Z+) Z· ( Z2)!

End(Z+n ) Zn Aut(Z+n ) Z·n (n N)!

End(Q+) Q Aut(Q+) Q·

" ( Aut(Zn) E ϕ(n) (n N)

. 2 # " , Aut(Zn) n ≤ 15

0 I P Zpn ! p E ! p > 2 n N! |Aut(P )| = (p − 1) α : x → x1+p (x P ) "

p Aut(P )

# I H E $# G! Aut(H) = {α|H | α Aut(G)}! # , H $

, G

$ ( Aut (Zp∞) E

( A Aut (Zp∞) p > 2 : p

. I . : Aut(Z2∞)

" # , x → x−1

: , E4 <

< ! Aut(E4) S3

G = x × y

( < " # : a, b G :

, ϕ G #! ϕ(x) = a ϕ(y) = b

. I |x| = |y|! , xmyn → ynxm+n m, n Z

4 ϕ G ! ϕ Aut(G) 2 #

ϕ, |x| = |y| = 4

m, n

N

( End(Z

+

+

+

+

) GLn(Z)

× . . . × Z

) Mn(Z)

Aut(Z

× . . . × Z

.

+

!

n

+

+

n

+

Mn(Zm)

End(Zm × . . . × Zm)

Aut(Zm

× . . . × Zm) GLn(Zm)

n

n

I E E : pn E Epn ! p E !

Aut(E) GLn(Zp) |Aut(E)| = (pn − 1)(pn − p) . . . (pn − pn−1).

2 # : , S3 )

< ! S3

> ( < g G

i(g) : x → g−1xg (x G)

, G

. I g G ϕ Aut(G) ! i(gϕ) = ϕ−1i(g)ϕ.

0 Inn(G) := {i(g) | g G} Aut(G) Inn(G) G/Z(G).

! ) ϕ : x → x−1 , G

! G

" 7 , " # # #

> 2 4 #

# I Aut(G) E $ ! G

$ 2 # G! Aut(G) = 1

H E G >

Aut(G)/CAut(G)(H) ≤ Aut(H)

> ϕ End(G) N E ϕ

G

( ) ϕˆ : gN → gϕN (g G) : ,

G/N ! ϕˆ : , ϕ

. I ϕ Aut(G)! ϕˆ Aut(G/N )

0 I N E G!

γ : ϕ → ϕˆ (ϕ Aut(G)) , Aut(G) Aut(G/N )

Ker(γ) = CAut(G)(G/N )

( I H ≤ G! , NG(H)/CG(H) ,

Aut(H) H G

. I G X! Aut(G) ! , " X/CX (G).

G = D2n (n ≥ 2)

( Aut (G) Z+n Z·n ! |Aut (G)| = nϕ(n)

. I n ! Aut (G) \ Inn (G) "$# 0 Aut (D4) D6

% Aut (D8) D8 & Aut (D∞) D∞

Aut(Q8) S4

) Aut(A4)

! I G E p G H Zp × Zp! |G : CG(H)| ≤ p

" I M N E G M ≥ N !G/CG(M/N ) ! , " Aut(M/N ).

# ( I G = a1, . . . , am ! |End(G)| ≤ |G|m

. < # G n |End(G)| ≤ nlog2n.

$ I G = A × B! (|A|, |B|) = 1! Aut(G) Aut(A) × Aut(B)

G , " n :

# H

> Aut(G) S "! Sn S ∩ Inn(G) = 1

I G = H1 × . . . × Hn! H1, . . . , Hn E

G!

Aut(G) Aut(H1) × . . . × Aut(Hn).

2 4

# , .