Белоногов. Задачник по теории групп

( G = (A1 × . . . × An) B1! A A1 ≤ G! B B1 ≤ G

{A1, . . . , An} = {A1}G H G = A1 B1 |{A1}G| = nD

.G A P B! P = RB,F E B

# 4 F n

A, B E P E , B SX

( A P B A × B |X| = 1

. I A! B X ! |A P B| = |A||X||B|

0 L # A! B! P! A P B

! G = Zn S2 Zn r Z2! n E 4 >

G Zn × D2n

" ( Z2 r Z2 D8

. ) Z2 r Z3 L 4

# G = Z4 S2

( G : x #! CG(x) = x

. 2 # Z(G)

$ m, n N! M E $GL(n, C)! m! S E $

GL(n, C) (S Sn) > M, S Zm Sn

G = Smn! m, n N > G !

, " Sm Sn

G E ! K E $ $# KG E

$ G K '5 > GL(n, KG) !

, " G Sn

G E n E >

Aut (G × . . . × G) ! , Aut (G) Sn

n

G A H ! |G : H| = 2 H = A × Ag! g G \ H g2 Z(H) > "$

t G \ H ! G = A t A S2

Z2 (1 2)(3) (Z2 S2) × Z2 (12)(3) S3

I A E B ≤ S(X, Y )

SX . Y ! -0/ !

A B (A BX ) × (A BY ),

BX := {b|X | b B} ≤ SX BY := {b|Y | b B} ≤ SY

! I A1 ≤ A! B1 ≤ B! P E , B SX P1 = P|B1!

A1 P1 B1 ≤ A P B

" I A1 ≤ A B1 ≤ B! A1 r B1 ≤ A r B

# G = A B! A E B ≤ Sn H

E : (H = A × . . . × A)

( H1 := {(a1, . . . , an) H |

1} B

a1

. .

. an =

n

H! ,

A × . . . × A

. I A ≤ A1

≤

A!

H

B

n−1

" ! ,

" (A × . . . × A) × A1

n−1

$ $

G SnO

¯

G = A rB! A = 1! H E G D := {f H | f (x) =

const x B}

(& .%

˜

˜

! ˜

˜

G = H B

H := iH (H)

B := iB (B)

˜

D := iH (D)

!

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( I

= 1

1 < N G

N ∩ H

.

˜ ˜

˜

CG(B) = D

Z(G) = Z(D)

0 I exp(A) = m exp(B) = n (m, n N)! exp(G) = mn

K˜ := iH (K) B˜

:= iB (B) > G = K˜ B˜

G = A rB! H E G! K := {f

H | b

B f (b) A }!

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A rB E p!

A¯rB E p : A

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C # CH ,$ α X

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γ H γ(xm1 1 ·. . .·xmn n ) = γ(x1)m1 ·. . .·γ(xn)mn =

α(x1)m1 · . . . · α(xn)mn x1, . . . , xn X m1, . . . , mn Z J

X : # #

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: X1 X−1H
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(xi, yj	1	m δ1	δn
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m + n! a−1 := x− n . . . x− 1				! 1a = a1 = a! 1−1	= 1!
		n	1
an := a . . . a n N ab = b−1ab

n

W E

X S = {xx−1, x−1x | x X} 3 a b X " W

H a W b! b a

# W S

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G # CRC!

# # (5 ./! (5 .( < W(X) X!

R(X) E X#!

WR := {uivi−1 | i I} R = {ui = vi | i I} R(X)

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A X! W : X a!

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W : X $! #
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W [a]W R [a]R 7 X W

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RR(X)! G X R (5 (8

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! 4

x 1 " x X >x −

x → [x] (x X) !

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x X : x : > F (X)

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X = {x1, . . . , xm} W = {w1, . . . , wn} X W

x1, . . . , xm w1, . . . , wn D " #

X R G = X | R G ! G = x1, . . . , xm | r1, . . . , rn

! x1, . . . , xm

I G X S ! S E # X!

! (X, S) !

G

I µ E ,$ ! # X! w E X! w(µ) ! " w #

: x X : xµ!

W (µ) := {w(µ) | w W } W W(X)!

R(µ) := {u(µ) = v(µ) | (u = v) R} R R(X)

	X E W E X
(	; [a]W [b]W = [ab]W	$
W : X

#

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4 # ( E X W

> G := X

( ) µ : x → [x] (x X)

X G

. A [u] (u W(X))

A ! |[u] ∩ (X1 X−1)| ≤ 1 u W(X)

0 G E "

Xµ = {[x] | x X}

F (X) E ! 4

! (5 . ( u¯ u W(X)

F (X)! u¯ = x11 · x22 · . . . · xnn ! u = x11 x22 . . . xnn (xi X, i = ±1)

( F (X) = {u¯ | u W(X)}! u¯ = v¯ [u] = [v]! u¯ · v¯ = uv

u, v W(X)

. ) u¯ → [u] (¯u F (X), u W(X))

4 # , F (X) X )

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% A "

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,

( I |X| = 1! F (X) E $

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W W(X)) > X W , ,

X !

γ: [u] → [u]W ( [u] X , u W(X) )

, X ) X W ! 4

Ker(γ) = {[u] | u [1]W } = {[w] | w W } X .

I n E $ n = 0! a an E $

|n| < : H

X W ! " (5 &

! a, b aba−1b−1 E $

" a, b a3, b2, aba−1 O

# R R(X)) G = X R > " X

u = v R G [u]R = [v]R

[u]W = [v]W ! W = WR

$ 7 a, b ab = bak, ba = abl ! k l E

!

R R(X) u, v W(X) ; H ( u = v R!

. [u]R = [v]R
u, v, s, t W(X)
( us = vt	# u = v s = t!
. u = u	" X
#

u, v, s, t W(X) ; H ( u = v s = t!

. [u]W = [v]W [s]W = [t]W " W W(X)

u, v, s, t W(X) ( u = v uv−1 = 1

. u = v v = u

0 u = 1 u−1 = 1

% u = v sut = svt & uv = t uss−1v = t 5 sus−1 = 1 u = 1

- uv = 1 vu = 1

R1 R2 E X#

( I R2 R1! X R1 R2 = X R1

. R1 R2 X R1 = X R2

G = X > R X

# ! G = X | R

! G = (X, ·) E > G = X | R ! R E

# xy = z! x, y, z X x · y = z

" G = X R E X#!

G ; H ( G = X | R !

. : g G X

w(g) !

g w(g) G!

w(x ) = x x X = ±1!

" g, h G w(g)w(h) = w(gh)

R

# R R(X)

( I G = X | R X G! G X R ,

ϕ:= ( X R → G | [u]R → uG (u W(X)),

uG E G u G! 4 #

, X R G

. I G = X R ! G = Xψ | R(ψ) ! ψ := (X → G | x →

[x]R)

$ G = X R ! R R(X) > "

µ X " H !

Rµ H! , γ G H #!

[x]γR = xµ x X R

G = X R

G = X | R R R(X)! µ E X

" H ; H

( µ , G H!

. R(µ) H : ! Cµ

R C! ! Cµ R C )

R G = X | R

	G = X | R E				I
				ϕ : X → G	!	G = X	ϕ
	R(ϕ)
			G! G = Xϕ | R(ϕ) ϕ
, G
	R {r} R(X) ; H

( r R!

." µ X " H

Rµ H ! rµ W H

! a2 = 1

# a4 = 1 ab = ba3

a, b abab−1 E

$

G = a Z3 (a G) ( G = a | a3 = 1

. G = a, a−1 | a3 = 1

0 G = a, a−1 | a3 = 1, ((a, 1), (a−1, 1)) = 1

aa−1 = 1!

# , .

((a, 1), (a, −1)) = 1

! > < G = X W H = X W W1 !

E

>

H G/N

!

˜

G

˜

W

W1

X

N = W1

W1

= {[w1]W | w1 W1}

" 7 a, b a3 = 1 , ! , "

Z3 × Z+

# m, n N

( a, b am, bn, a−1b−1ab Zm × Zn