Белоногов. Задачник по теории групп

! # : An

n ≤ 10O

! n ≥ 3 An # $

Sn

! k N! k 3 ≤ k ≤ n > An

$ k Sn

! " n ≥ 3

( An = (123), (124), . . . , (12n) !

. An = (123), (123 . . . n) 4 n! 0 An = (123), (23 . . . n) 4 n

! " n > 2 An ! , "

Sn−2

! ! f Sn (i, j) !

1 ≤ i < j ≤ n f (i) > f (j). I(f ) E # f. >

f 4 I(f ) 4.

! " ) # (1 k) Sn (1 <

k ≤ n)
	! # Y X SX (Y ) := {g SX | yg Y y Y }
Y SX N := {g SX | yg = y								y
Y } Y SX > N SX (Y )								≤ SX
SX (Y )/N SY									. .
	!$ {X1, . . . , Xk} E			k			X! X = X1
									. . . Xk
(k		N) S(X1, . . . , Xk) :=		i	SX (Xi)! SX (Xi) E
				=1
# Xi SX -.8 !

SX ! " # "

( S(X1, . . . , Xk) SX1 × . . . × SXk .

. SX ! 4 # SX !

# SX

0 n = n1 + . . . + nk, n1, . . . , nk N.

Sn, , Sn1 × . . . × Snk ?

! < SX ! |X| < ∞! SX

! " # $#

+ ! a g E : SX

a = (x11 . . . x1m1 ) (x21 . . . x2m2 ) . . . (xk1 . . . xkmk ),

ag = (xg11 . . . xg1m1) (xg21 . . . xg2m2) . . . (xgk1 . . . xgkmk ).

! g Sn m Z > : g gm Sn

(o (g), m) = 1

! n ≥ 4 $ " # "$ Sn

# "$

! ( A Sn # : 4

: < $ a = (1 . . . m) : g #! ag = a−1.

. n ≥ 4 # # : g Sn

: # 2o(g);

"$#

! @ "$ (1 2) Sn (n ≥ 2)

# S({1, 2}, {3, . . . , n}) = Sn({1, 2}) -.8

! n ≥ 3 Z(Sn) = 1

! ! I a E $ n Sn!

CSn (a) = a |aSn | = (n − 1)!.

! " a b E $ Sn ! ab = ba > a b

! a = b

! # ( A S4 # $ "$ (12)(34).

. 2 # |3Sn (a)|, a = (12)(34) . . . (n − 1, n)! " 4 n

!$ g E (z1, . . . , zs) Sn < " i = 1, 2, . . . , n c(i) i

(z1, . . . , zs) >

n

|CSn (g)| = c (i)! ic (i)

i=1

! ! |gG| = n! / n c (i)! ic (i).

i=1

! 2 # 4 : S4 S5

! :

! a E $ (z1, . . . , zm) An ( aSn An! 4 aSn = aAn ! aSn E B

4 : An

. aSn = aAn ! zi 4

! x y E $ k An! k ≤ n − 2 n ≥ 5 > x y An

! 2 # 4 : A4 A5

! ( 7 A5

. 7 S5 " " !

E A5

! N n N n ≥ 5 ( 7 An

. An E Sn

! ! A A5

( "$ D

. $ "$ , Z2 × Z2D

0 %! " $

! " 7 A5 : a b!

"

a5 = 1, b3 = 1, (ab)2 = 1.

! # A A5

( # 4 : !

.

!$ A A6

( "$ D

. $ "$ , D8D

0 $ %! "

$

! n An "$ O

! A A7 # B A! A S4 B Z3!

A AB

! f (x1, . . . , xn) E

X = {x1, . . . , xn} :,,$! S E

γ SX ! f (γ(x1), . . . , γ(xn)) = f (x1, . . . , xn) > (S) E SX

f (x1, . . . , xn) ) 2 # # "

n E # " H

x21 + 2x2 + x23 D

(x1 + x2)(x1 + x3)(x2 + x3) D (x1 − x2)(x1 − x3)(x2 − x3)D x1x2 + x3x4 D

x1x2 − x3x4D

(x1 + x2)(x3 + x4)D

(x1 + x2 − x3 − x4)2D

(x1 − x2)(x1 − x3)(x1 − x4)(x2 − x3)(x2 − x4)(x3 − x4)D

(x1x2 − x3x4)2 + (x1x3 − x2x4)2 D (x1x2 − x3x4)2 + (x1x3 + x2x4)2D

x21x2 + x3x24 + x21x3 + x2x24 D x21 + 2x2 + x23 + 2x4;

(x1 − x2)(x2 − x3)(x3 − x4)(x4 − x1)D x1x3 + x2x4x5

! " n # x1x22 + x2x2 + x3x2 + x4x2 + . . . + x x2 n.

3 4 5 n 1 $

- . %

, X E : #

SX '( |X|

: # SX : X

"

G ≤ SX ! x X Y X H

Gx := {g G | xg = x} E x GD

GY := {g G | yg = y y Y } E

Y GD

G(Y ) := {g G | yg Y y Y } E

Y G : E G 6 ( D

xG := {xg | g G} E G! x !

G " B 4

XD

F ix(g) := {x X | xg = x} E &

gD

F ix(H) := {x X | xh = x h H} H SX

) & x y G Gx ∩

Gy. 7 G !

" x, y X g G #! xg = y 2 " " Y E

X 7 ! G Y ! Y

B G

G ≤ SX 7 G X!

Gx = 1 " x X < " k N G

k k !

" # (x1, . . . , xk) (y1, . . . , yk)! k : X! : g G #! xgi = yi

i = 1, . . . , k

S := {X1, . . . , Xm} E X! X = X1 . . . . . Xm

7 ! S G X!

Y g S Y S g G 3 {{x} | x

X} {X} " 7

! #

# ! > !

G E P E , G

" SX A : ! P

G X ! G

X P 3 C PC " !

! P 4 D : P(G) "

G |X| P

P ! Ker(P) = 1! !

	P(G)		!
		+ G X
		X ×G X (x, g) → xP(g)		P E

G → SX 3 " #

"

X E :

G H ≤ G.

( I " h H P(h) : x → xh (x X)

X Xh = X! P : h → P(h) (h H) , H SX 6 .& A :

! P H X !

H X

. I " h H Q(h) : x → xh (x X)

X Xh = X! Q : h → Q(h) (h H) , H SX 6 .& A :

! Q H X !

H X

G ≤ SX H ≤ SY 7 ! G H

! "

f X Y , ϕ G H !

f (xg) = f (x)ϕ(g) x X g G

G H , # " C C

A : G → SX B : G → SY "

! fX Y ! f −1A(g)f = B(g) g G

W E ( ! . 0

: W !

!

W J # W $#
#	# 6 %% )
n! n E W !
E(n) F E	, "
W J # W ! "
, F # E(n) 6 %% )
F W

" G E X! x X. Y X ( Gx x G! GY # Y G

G(Y ) # Y G E G

. Gx = 1

x X

0 GY G(Y )! G(Y )/GY , SY I G = SX ,

G(Y )/GY SY

" > G ≤ SX x X > ( |xG| = |G : Gx|D

. I G ! |G| = |X| · |Gx|D 0 g G, Gxg = g−1Gxg

	" G ≤ Sn g E p: G
p! n > p |F ix(g)|
	" G E X! x
X!	H ≤ G
	( H X GxH = G
	. H GxH = G Gx ∩ H = 1
	0 I H ! |H| = |X|
	" G ≤ SX , N G N X > G = N Gx
	x X

" G E SX ( G

. G E SX

" ! @ #

" " I A E $			Sn! A = a ,
a E $ n
" # G E			SX x X
> NG(Gx) # F ix(Gx)
"$	Sn
n − 1 ! "
" = F # I G ≤ SX			t E !
t|G| =		g
		|F ix(g)|.

G

" G E ! " k

m1, . . . , mk E : >

H.O.K(m1, . . . , mk) ≤ |G| ≤ m1 . . . mk,

m1, . . . , mk

" G E SX x X

( I Gx < H < G B := xH ! {Bg | g G}

G

. G Gx E G

" H E # G > H ! ! "

4 # Hg (g G) !

!

" p E > "

Sp

" G = (1 3), (1 2 3 4) E S4 !

G 4

" ! L " 4 ! 6 (%

! S4

" " ( 7 Sn n

. 7 An (n − 2)

0 L (n − 1) Sn

" # G E SX ! x X k N ; H

( G (k + 1)!

. Gx # k X \ {x}

"$ I G E k Sn! |G|

n(n − 1) . . . (n − k + 1)

" L (n − 2) Sn

" .

" I G ≤ SX ! G 1 < N G! N

" G ≤ SX H ≤ SY ; H ( G HD

.f X Y !

g → f −1gf (g G)

G H

" < ( . ! ,

: ,

" > G = (X, ·) E

( " g G Rg := (X → X | x → xg)

!

RG := (G → SX | g → Rg)