Белоногов. Задачник по теории групп

2 # Hom(G, H) ! (G, H)

(Z+, Q+)!

(Zn, Q+)!

(Q+, Zn) (n N )

( I ϕ E , Q+ C+! ϕ(q) = qϕ(1)

q Q+

. 2 # Hom(Q+, Z+) Hom(Q+, Q+)

$ % & #

a b G " G!

: g G #! g−1ag = b g−1ag

ag J

aG := {ag | g G}

& G

: a =" # : ! !

4 4 : A ! # G

H

|G| = |C1| + · · · + |Ck|,

C1, . . . , Ck E 4 : G 3

"

)H " # # G "

p p !

# p! |G|

: # & (-

) B " 4 :

G G ) !

X G #H Xg = gX g G NG(X) = G U * W Q * W! * H ≤ G H E

G! H G. I H ≤ G,

HG := ∩ Hg H G

g G

< X Y : a G " H

Xa := {xa | x X} E ! X GD

{X}G := {Xg | g G} E &

G # XD

XY := {xy | x X, y Y }

< " g G x → xg (x G) ,

G G

A B E G g G

( (A B)g = Ag Bg

. (A ∩ B)g = Ag ∩ Bg 0 (A \ B)g = Ag \ Bg

% (AB)g = AgBg

& (A−1)g = (Ag)−1

5 A g = Ag

- NB (A)g = NBg (Ag) 6 CB (A)g = CBg (Ag)

8 I A ≤ B ≤ G! |Bg : Ag| = |B : A|.

I A B E G!

A B! A ∩ B! A \ B! AB, A−1! A ! NB (A)! CB (A)

) B 4 :

4 # #

> G E ! a G X G >

( |aG| = |G : CG(a)|!

. |{X}G| = |G : NG(X)|

) 4 : #

D6, D8, Q8

! a1, b1, a2, b2 E : G ! a1 4b1 a2 4 b2 F a1a2 4 b1b2?

" ! " 4

: O 3 & %0

# G E ! # B "

4 : ! (!

# # GO

$ G E d E # #

4 : G! !

: > d |Z(G)|

I G E ! # #

4 4 : G \ Z(G).

G E pn! p E n N ( Z(G) > 1

. I M E G! M G |M| = pn−1

0 I H < G! H < NG(H)

7 p2! p E " !

$ ! $

p

G E p3, p E

2 #

( |Z(G)|,

. 4 : G.

! # G

r # (x, y) xy = yx G J #

4 : GO

I N G |N | < ∞! |G : CG(N )| < ∞

! < " 3 ! "

G E p !

( H < G! p |G : H|D

. G pD

0 p |Z(G)|D

% . 0 0 6

" G E p E I

p P G G! P

p: G ! ! P

p G

# G E			! " 0 4
: > G Z3, G S3.
$ G E			! ! "
4 A B CA B C CA ≤ BC G O
G E			n. ; H
(	"	m n xm = 1
m # GD
.	"	m n G
mD
0 G $
I H ≤ G! HG (=g∩G				Hg) G I |G : H| < ∞!

|G : HG| < ∞

I A B E G! AG ∩ BG = (A ∩ B)G

H E π G > G

π# |G : H| E π

! π

L . &% ! 4 C

C C C ! H K E

# G! G

S ! G = H ◦ S = S ◦ K

B

! 4 # : #

! ! #

#! $# # pα! p E α N {0}

" I G E G = aG1 aG2 . . . aGn , G =

a1, a2, . . . , an a1, . . . , an E : G).

# G E ! " # #

# G ? <

$ ) ! " 4

4

G

! 4 ( G O

G E H < G.

( I G \H 4 : G, H E

4 G D(H).

. G\H 4 :

G. G O

H ≤ G |G : H| = n < ∞.

( < " g G gG ∩H B n

4 : H.

. I G k 4 :

! H 4

: : k|G : H|

A E 2 G G

% 0 > " a A aG = aA! aG = aA bA,

b A, aA = bA |aA| = |bA|.

ϕ E , G! g G! A B E

G

( (gG)ϕ = (gϕ)Gϕ
. I A B G!	Aϕ Bϕ Gϕ
0 A ! A B	ϕ! !

.

G E H G

( " , ϕ G

|{Hϕ}Gϕ | |{H}G|.

. I H ≤ G

N G!

|{H ∩ N }N | |{H}G|.

! G E

! N G, g G,

G

= G/N

= gN

g

(

G

G

G

gN

|g

| = |g

|/|g

∩

G

|

. C

(

) =

|CG(g)||g

∩ gN |

g

G

|CG(g)|

|N |

0

C

(

)

C

(g)

g

|CN (g)| ≤ |

G

| ≤ | G

|

" G E

! ϕ E 4 , N

g G

( I g N \ {1}! |CGϕ (gϕ)| < |CG(g)|

. I CN (g) = 1, CGϕ (gϕ) = CG(g)ϕ CG(g).

# H G ϕ E # , G G/CG(H) ( I Z(H) = 1! CGϕ (Hϕ) = 1 Hϕ H

. I G ! Hϕ H ! Z(H) = 1

$ A, B E G A B

( AG BG BG \ AG (B \ A)G

"

. ; H

(B \ A)G = BG \ AG!

AG ∩ B A : ! A B

G

A B E G!

#! g NG(A B) >

	g NG(A) ∩ NG(B),
	Ag = B Bg = A.
|NG(A B) : NG(A) ∩ NG(B)|?
G = a, b (a, b G)
( I a			G! b−1ab = as $ s.

. I : a b #!

! (

a b E : ! b−1ab = a2 o(b) = 5 o(a)O

b−1ab = as! a b E :

α β! ! s E $

( s(β,ϕ(α)) ≡ 1 (mod α)

. (β, ϕ(α)) = 1 ab = ba

ϕ E ,$ #

G E p E #

4 :

( p G 4 G #

"! #

. p G

Z(G)

@ . E

! " 4 :

! a E # : G > ( aG ∩ a = aNG( a )!

. |aG ∩ a | |aG| ! 0 |aG ∩ a | ϕ(o(a))

" a G, o(a) = n |aG| = j (n, j N) I (j, ϕ(n)) = 1!

: a G

# t E "$ G, CG(t) = t × H, H S3,Q := t × a , a E "$ H > |NG(Q)| = 4m,

m {1, 2, 3}.

$ A ≤ B ≤ G B ! B

! 4 " A G! B H E

B j E # #

|A : A ∩ A1|! A1 H. > |B : A| j

A B E # G! α E

! 4 A B! β E !

4 B A >

α|NG(A)| = β|NG(B)|.

I H ≤ G, g G gH E G!

H G

I M E G! {g G | gM M} E G {g G | gM = M} G.

I B G

# 4 #

G! G

H E G I

M2 := MM M G H

J G!

H JJ−1 < G.

A ! J {1}! G

M E " G H ≤ NG(M) >

M \ H M, H = M H.

! G = M ! M E G! H E G > M \ H G

" M E G

( M \ NM (M) M

. 2 NM (M) M

# M E 4 : G G = M ( G = M \ H " # # H G

. I M = M1 M2, G = M1 , G = M2 .

$ G E M E 4

" G = M ! G = M1 M1 M) > NM (M) =

M E	G n N >
|M : Mn|	n Mn := M . . . M

a b E : G ) (a1, b1)

: G !

a1b1 = ab b1a1 = ba.

A ! (a1, b1) |CG(ab)| ?

< : a, b, a1, b1 G H

( a1b1 4 G ab! b1a1 4 G baD

. " : x y G ! a1 = x−1ay b1 = y−1bx

	D	{a, b} G ; H
( a2 = b2
. b = ca! c CG(a2) ca = c−1
	G = a, b ! a, b				G ba = b−1	= b >	a2		Z(G)
G/ a2 E :								α	β
	( A GL2(Q) G $
								0	1

(α, β Q, α = 0) #

. A G # : a! # 4 G " # #

" am m = 0

0 A G # ! G " # #

# #

! A #

# # #

" A ≤ B ≤ G, g G Ag B. F |B : Ag| = |B :

A|?