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Гомельский государственный университет им. Франциска Скорины

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Белоногов. Задачник по теории групп

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A "

m, n E !

p E ! " '(8! q E ! p

§1

# ; 2 $" ·! Φ! 4

C C,$! ,$ ϕ ! (M, ϕ) E

! # #H

( ϕ(ϕ(x, y), z) = ϕ(x, ϕ(y, z)) x, y, z M!

. e M #! ϕ(e, x) = x x M!

0 x M y M #! ϕ(x, y) = e > ,$ Φ H ! ! ,$ ϕ(x, y) = xy

# 2 " # # ,$ ϕ (

(M, ϕ) G

H < ! H ! H ! H ! H ! H ! 4 H

. H X = A∆B

! . H {(a, 1) | a R} {(0, b) | b R} \ {0}

0 H {(a, −1) | a R} E "$# G G

: !

$ Mλ E λ < 0

. H <

§2

" % H M E G M = ap ! p E

A Z4 E 0 ! Z2 ×Z2 E & ! Z6 E % !

S3 E &

{n | n N} E Z+! |Z+ : n| = n

M E Z+ M = p ! p E

# , Z+

!	. H 3 #	{1, g, . . . , gd−1}!
d = \|m\|! o(g) = ∞! d = (\|m\|, o(g))! o(g) < ∞
"	0 H H G = S3, \|H\| = 2D	H G = S3. \|H\| = 3

0 H k = 1 G , Z2 A B A B

Z2 × Z2 k = 2 G , Z3! Z4! Z2 × Z2! Z6 S3!

$ 6

H G = 1 G ZpD H G Zp2 D

H G , G Zp3 G ZpqD H G , G Zp4 Z2 × Z2

! H G Zpm D H G Zpmqn

$ . H o(−1) = 2! o(i) = o(j) = o(k) = 4 „M* H {1}! i !

j ! k ! Q8

# $" n

$ G T " # G

: E "$ |T | = 2

"	A ! 4
$	Q+

§3

0 H : a a−1 o(a) = ∞D : ak (k, o(a)) = 1

o(a) < ∞

& H o(a) = ∞ E : 4 aD 4 o(a)a D o(a) = 2k "

{a, a3, . . . , a2k−1}

"	0 H Z30 = Z2 Z3 Z5 D Z61 = Z61 D Z10 ×Z60 = ABC ! A Z2 ×Z4 !
B Z3 ! C Z5 × Z5				4 ,	1			4 (
	. H '1, 1	2	3	4 ,	1	2	3	4 (
	2	3	1	4	3	1	2	4
	A H ( ! 0 ! % H % ! H ( ! %
		(pn − 1)(pn−1 − 1) . . . (pn−m+1 − 1) .
		(pn − 1)(pn−1 − 1) . . . (pn−m+1 − 1) .
		(pm − 1)(pm−1 − 1) . . . (p − 1)

#	2
	CG(x) =	' a			0
			0		d
	CG(y) = ' a
			a		0
	CG(z) =		a		0
	CG(z) =			b	c

					e
			d		e


	. H

(

| a, d C \ {0} C· × C·.

b	\|	a	C·		, b		C( .
a	\|
0		M3		3(C)
0		M3	×	3(C)		\|	ac = 0 .
			×			\|
a
a

−1 0

, 1

−1

1 aa2

−

a, b, c C, b = 0

π |π| = 2

" G E $

|Z(D2(2n−1))| = 1! |Z(D4n)| = 2 n > 1! Z(D∞) = 1

! 6D 5

G Zp |G| = 4

G E $

G E $ ! : : .

; G = a1, . . . , an G = M ! M G

> " i {1, . . . , n} ai

: M M−1! ai Mi ! Mi M |Mi| < ∞ :

G M1 , . . . , Mn = M1 . . . Mn

! A # : #

% H |Q(p) : kQ(p)| = k & H M E Q(p)

M = qQ(p)! qE q = p

" o((a, b)) = [o(a).o(b)]

§4

G = D6 M = {a, b}! o(a) = 2 o(b) = 3

# )

; E4 D8

# . H (! Z2 D2m! m | n

$ A , H (! Z2! D2m m > 1 D∞

( H Z2 × D8, D16

D(Z2∞)

SLn(C)

§5

Z 4 : %! &! &!

" 7 .

# |G| = 2

( H p! . H p2 + p − 1

< ! : r/|G|

# + # & & (

$ G E $

# |G| p2 pq

$ G % 4 G Zp3

|G| = pq

G , E4 Zpqr! p, q, rE

. H G D(H)

$ ( H A G = S3

+ ≤ 2

# ; CG(t) = b Q! H = b a x NG(Q) > tx tNG(Q) {t, a, ta}! & & ( m := |tNG(Q)| = |NG(Q) :

CNG(Q)(t)| = |NG(Q) : NG(Q) ∩ CG(t)| = |NG(Q) : Q|! |NG(Q)| =

|Q|m = 4m

" ( H A G = S3

	<					! g =	2		−1! A =
" 2 H G = GL2(Q)! b = 1					1	! g =	2	0	−1! A =	b2 ! B =	b
> 1 = \|B : Ag\| = \|B : A\| = 2				0	1		0	1
> 1 = \|B : Ag\| = \|B : A\| = 2							−2		0
#	G :=	a, b ! a =	1	1		b =	−2		0
			0	1			0		2
!$	CG(t) = A2 × t ! A2 := {a A \| a2 = 1} Z(G) A2
A2 < A G A2 = A
!	H n = 2k + 1H {1}, {a, a−1}, {ak, a−k}, a b n = 2kH

{1}, {a, a−1}, {a(k−1), a−(k−1)}, a2 b! a2 ab

H 4 H ( 4

nH am m|n G ! am b m|n m = 1

D . H 4 nH am m|n! a2 b ! a2 ab G

! am b am ab m|n m > 2

!	H 3 &-. H G = D2pn ! p E 4
!	H n = 2(2k − 1), k N ! H n 4
!	( {1}! {am, a−m} m N! a2 b! a2 ab
. {1}! am m N! a2 b ! a2 ab G
	§6
	. H kZ M k N {0} 0 H {0}! {1}! {1, −1} (= Z·)!
	Z6[x] x2 + x = (x − 0)(x + 1) = (x − 2)(x − 3) = (x − 0)(x − 2)
	K n : a1, . . . , an < a K \ {0}

aa1, . . . , aan a1a, . . . , ana!

$ K

$ ; H ≤ F · |H| < ∞ > H !

0 6 % H : h n ! : H

" " xn = 1 3 ! 5 (/ |H| ≤ n 2 H h n ! ! H = h

' a

a, b

H ' A

−b

CM2(K)(x)(

A, B, C, D

Diagn(C)

a b

c d 0

GL2(C), e C

0 0

Diag

n(K)

0 H ({0, 2}, ◦) K = Z (Q \ {1}, ◦) K = Q

·∩ (1 + Ker(ϕ))

= {[0]n, [m]n, [2m]n, . . . , [(k − 1)m]n}! k =

Ker(ϕ) = mZn

n/m

GLn(R) ∩ (e + Mn(I))! e E

N = e + M2(I)! e E

$ I = pZp2

: ˜

(0, 0)

:= (0, 1)

b := (1, 0)

˜ ˜

+ H

1 := (1, 1)

{0}

{0, a}

{0, b}

{0, 1}

˜ !

{0}

{0, a}

{0, b}

§7

! Z $ .! 0 % 8! % &!

! 5 n {5, 6}! (. n = 7! (& n = 8! ./ n = 9! 0/

n = 10

! (n − 1)!

! ! ak (m, k) $ #

! (a1 . . . akb1 . . . bl)

! " 2k − 3

! # . H (n/2)!2n/2

! # ( H & 4 : : &

" .

. H A A5 E 0 4 H %

! , A4D (/ ! , D6D 5 ! ,

D10

! 4 ≤ n ≤ 7

! (1234)(56), (12)(56), (567)

§8

" 2 H (! .! 0!

! S3D H $

% D

H A4D H .

" ! 0 H S4

" ( H {1, α} Z2! α E #

. H {1, β} Z2! β E [a, b]

0 H (
" "	( H Z2 . H ) E
"	7 , ! H S3! Z2! D8! E4!
E4! E4! (! E4! Z2 × D(T )! Z2 × D(T ) T E # ! Z2
	§9
#	( H Zpn ! n N . H Z6 0 H E4
#	Z ,$# Q8 E 0! D8 E ((
#	( {Zk, Zl}! k, l E n [k, l] = n . Zk!
k\|n (k, n/k) = 1
#	D6 = a 3 b 2 D D10 = a 5 b 2 D D30 = a 15 b 2 = a3 ( a5 b ) =
a5 ( a3 b ) ( a5 b D6! a3 b D10 )
#	2 H G = E4
#	G = D6
#$	A G = E4
#	n 4
#	D∞
	§10
$	( H ; S3 . H 2
$#	; # S

Sylp(G)

$ %6

$ % H .! G E $ ! 1 + q! G E $ !

$# G , # H Zp, Z4, Z6, Z9, E4, E9, S3$ Q H 5! H .(

$# n1n2

$ ; D8 × Z2

§11

( 0 & 5 H ϕ E : , ϕ E ,

0 5 % H ϕ E : , x = 1

( ! . ! % ! & H ϕ E : , ϕ E , ( ! %

& x = 0

"	. H 1! Z2! Z2! Z4! Z2! Z6! E4! Z6! Z4
	5
$	\|G\| ≤ 2
	S4
	+ # ( %0

§12

= 3!

= 2! D4n

Zn! D2(2

n+1)

Z2n+1 !

D∞ : D

= 4

D∞ Z+ | A|

" #| |

∞|

# :

A4 E4!

S4 = A4

SL2(Z2)

Z3! SL2(Z3)

; ) µ !

: g g = ϕ(g1, . . . , gn)

! . H 2 ; D16

. H 2 ; Q8

§13

G = A4 : Li(G) = G E4 i ≥ 2! G = 1! Zi(G) = 1 i ≥ 1D G D 1 < G < G E #

$# # # G

	G = S4 : G = A			! G	E	! G = 1! Z (G) = 1	i	≥	1! L	(G) =
G			4		4	i		≥	i
G			i ≥ 2D GD G D 1 < G < G < G E
# $# # #
	Q8	0 $ ! E
	D8	- $ ! 0 E

G = a γ b 2! γ N {∞}! ab = a−1 A

G = a2 ! G = 1

( I γ E 4 ! Zi(G) = 1

i ≥ 1!

Li(G) =

= a

≥

2k−i

1 ≤ i ≤ k − 1!

. I γ = 2

(k i

1N)! Zi(G) = a

Zk(G) = GD

Li(G) = a2 −

i ≥ 2 = 1 i ≥ k + 1D

0 I

γ = 2

m, k ≥ 1, m

4 !

m > 1

−

Zi(G) = i 1

1 ≤ i ≤ k!

Li(G) = a

2 −

Zi = ak

= H(G) i ≥ kD

2 ≤ i ≤ k + 1! Li(G) = a2

i ≥ k + 1

2i−1

% I γ = ∞! G D∞! Zi(G) = 1 i ≥ 1D Li(G) = a

i ≥ 2

! G = A! G = 1! Zi(G) = ai i ≥ 1! H(A) = Li(G) = A

i ≥ 2

3 -%5

§14

( H ni := ni(G)

< G = Z2 × Z4

n2 = 3! n4 = 4

G = Z2 ×Z2 ×Z4 n2 = 7! n4 = 8 < G = Z3 ×Z4 n2 = 1! n4 = 2! n3 = 2!

n6 = 2! n12 = 4 < G = Zp × Zp2 × Zp3 np = p3 − 1! np2 = p3(p2 − 1)! np3 = p5(p − 1) < G = Zpm × Zpn m ≤ n npi = p2i−2(p2 − 1)

1 ≤ i ≤ m m < n npm+j = p2m+j−1(p − 1) 1 ≤ j ≤ n − m

. H Z $ i G ni(G)/ϕ(i)

. H 2 ; E4

; E4

G E A B! A × B

H A = a B = b G = {(1, 1), (a, b), (a2, 1), (a3, b) Z4} H

Z4 × Z3 H G H G H >

, Z4 × Z2

G1, . . . , Gr−m A = Gr−m+1 ×. . . ×Gn

> " G

G1, . . . , Gr−m A

2 H ! ! E

§15

# ( Z+ ×Z+D . G = a b ! a×b2 Z+ ×Z+ G/ b2 D∞

$ ( H Z2! . H D6! 0 H S4D % H D∞

0 H G A B A = 1! Ker(P) =

B	. H G = ( a1 2 × a2 2 × a3 2) ( b 3 c 2)!
"	. H G = ( a1 2 × a2 2 × a3 2) ( b 3 c 2)!
3	. G , D8 × Z2

#	. H G = ( a 4 × b 4) c 2! ac = b Z(G) = ab
#	Z(G) = {((a, . . . , a), 1) \| a Z(G)} Z(G)

		n
	§16

b c S3

" Z3

$ 2 ; , G = Z6 Z2

. H , a 4 ϕ b 4! ϕ(b) : a → a−1

G Zp S2

# |G| = 12! |H| = 24

$ . H ./

( H G/N , # " H E4 H

D2m! H D∞!	H D∞
	§17
! ( H	a4 ! a2 ! a2
. H A : P \ a " %		4
. H A : P \ a " %		0 H	a !
: P H {1}! {a4}! {a2, a6}! {a, a7}! {a3, a5}! a2 b! a2 b3			a !

a2 b Q8! a2 ab Q8

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!	. H 3 #	{1, g, . . . , gd−1}!
d = \|m\|! o(g) = ∞! d = (\|m\|, o(g))! o(g) < ∞
"	0 H H G = S3, \|H\| = 2D	H G = S3. \|H\| = 3