Белоногов. Задачник по теории групп

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: x−1gx! g, x G! g G 3 " 4 : 4 '&

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X = {1, x1 . . . xs | xi X X−1, s = 1, 2, . . .}.

7 " (

E $

( n N, g G o(g) = n. I m N (m, n) = 1,

t N ! gmt = g ! ! mt ≡ 1(mod n).

. m, n N. > " $ x y !

(m, n) = xm + yn.

G = a E $ a G H ≤ G

( I H = am1 , am2 , . . . ! {m1, m2, · · ·} Z! H = ad ! d

# # m1, m2, . . . .

. ak ∩ al = a[k,l] " k, l Z! [k, l] = |kl|/(k, l)

k l

0 L : ! # G G !

|G| = n, : ϕ(n)! ϕ E ,$ # ϕ(n) = |{m N | m ≤ n, (m, n) = 1}|

% I |G : H| = m! # " # : H

m# " " : G & A : G! "

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4 π:

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! a E : # G!

mn! (m, n) = 1 > G : g, h

!

a = gh = hg, o(g) = m, o(h) = n,

4 g h " : a a = g h

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( I |G| = mn! (m, n) = 1! G

M m N n : G = M N

. I |G| = pa11 . . . pass ! p1, . . . , ps E

ai N! G = P1 . . . Ps! Pi E pai i

pi G (i = 1, . . . , s)

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Pi . G {Z30, Z61, Z10 × Z60}

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$ 7 S3				: a b! "
a2 = 1, b3 = 1, (ab)2 = 1.
A S4 :
a = 1	2	3	4 , b = 1		2	3	4 .
2	3	4	1	4	3	2	1

( a b E . S4

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0 7 S4 : a b! "

a4 = 1, b3 = 1, (ab)2 = 1.

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G G " # G O

( g → g−1

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0 g → ag

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(g G) (g G)

(g G) , a G ga (g G) , a G

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g = xax , x G.

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CG(X) := {g G | xg = gx							x X} G A
! CG(a) Z(G) E G
# J Z(G) # # GO
$ I H ≤ G!					CG(CG(CG(H))) = CG(H)
2 # $ : H
z =		1	0	y =	1	−1 GL2(C)!
		0	−1		0	1
		1	0 0
z =		0	1 0	GL3(C)
		1	0 1
A B E G >
						b	|A ∩ CG(b)|.
				|B	∩ CG(a)| =
				a A		B

ϕ E , G H

( Z(G)ϕ = Z(H)

. CG(X)ϕ = CH (Xϕ) " X G

( 7 SL2(C) "$"

. 2 # "$ GL2(C)

a b E : G

( o(a) = o(a−1)

. I b = g−1ag g G b 4 a G!

o(a) = o(b)

0 o(ab) = o(ba)

% I G "$" t, t Z(G).

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! " abc #

#! "

. < : a, b, c o(abc) = o(cba). ; S4

! G = a, b , a−1ba = b−1 b−1ab = a−1 > a4 = b4 = 1. I : a = 1 = b, G , Z2 × Z2 Q8

" I x, y E : G

! x−1yx = y−1, : x y 4 #

# G = a, b bab = a! a2 Z(G).
$ I a, b G! a4 = b4					= 1		ba = a2b2!	ab = (ba)2
(ba)5 = 1
I a, b G ba = a2b2!
( ba2 = a2n(ba2)b2n " $ n D
. a2 b2 " !
0 ab4 a4b "
I a b E : ! ba = a4b3, o(a4b) =
o(a2b3).
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	"
ab−1
I a, b G! a6 = 1					ba = a4b! a3 = 1			ab = ba.

a b E : n N.

( I an = bn, [n, o(a)] = [n, o(b)].

. I o(a) = pα o(b) = pβ , p E α, β N,

an = bn = 1, o(a) = o(b).

I A B E G B A, B \ A = B.

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H < G G \H "$# > H

G E : . !

|G : H| = 2 G D(H)

G := D(Zn)! n N n ≥ 2

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2n.)

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"$#

0 D4 Z2 × Z2.

% n > 2 D2n

& D8 Q8

D∞ := D(Z+). A ! ,

D∞,

> G " $" a .

G \ a "$#

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a, b, x, y GH
( G = a, b , o(a) = o(b) = 2 o(ab) = nD						D
. G =	x, y , o(x) = 2, o(y) = n, x = y x−1yx = y−1
0 G D2n
U "		G H
( G = a, b , o(a) = o(b) = 2 o(ab) = ∞D					D
. G =	x, y ! o(x) = 2, o(y) =		∞	x−1yx = y−1

0 G D∞.

2 # $ D2n D∞.
! 3 " S4 " : H
(		1	2	3	4		,		1	2	3	4	D
		2	1	3	4				3	4	1	2
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		2	1	3	4				2	4	3	1

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# I G E 2p! p E ! G

, Z2p D2p

$ 3 " ! "$O

7 Q8 . .8 : i j! H

i4 = 1, j4 = 1, i2 = j2, ij = j−1i.

G = x, y ! : x y "

H

x4 = 1, y4 = 1, x2 = y2, xy = y−1x.

> " # GH ( x = y = 1 G = 1D

. x = 1, o(y) = 2 G = y Z2D 0 o(x) = 2, y = 1 G = x Z2D % o(x) = o(y) = 2 G D4D

% o(x) = o(y) = 4 G Q8

G ! "

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0 7 Q	+ : 1
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			pn

n N

% Q+ = 2!1 , 3!1 , . . . , n1! , . . .

< " p Q(p)

p #! pmn ! m n E $

( Z+ < Q(p) < Q+

. Q(p) = 1p , p12 , . . . , p1n , . . .

0 I H Q(p)			a		b
			pi pi+j !
a, b, i, j N! H	d	! d = (a, b)
	pi+j
% ) Q(p) !					4

$ kQ(p) (= kp , pk2 , . . . , pkn , . . . )!

k N (k, p) = 1! , # Q(p)

|Q(p) : kQ(p)|O

& L Q(p) 5 Q(p) Q(q) p = q (q E

a b E : #

:

( I an = bn! n N! a = b

. I an = bm! {m, n} N! a, b E $

G E !

# # # > G E $

G : a b m n ! ab = ba

( I (m, n) = 1! o(ab) = mn a, b = ab

. I (m, n) = d > 1! o(ab)	mn	mn
	d	d2

0 ! . o(ab) mn2 !
mn			mn	d
				mn
		d !	d2 !	d
% I a	b	= 1, o(ab) = mn
	∩	: mn
		d
& A a, b			d
5 I m = pα n = pβ ! p E α < β! o(ab) = pβ

a E : ! H E #

a H! #

o(a; H)! " H # an (n ≥ 1) : a H ar E !

o(a; H) = rD an (n ≥ 1) H! o(a; H) = ∞

< " H ( o(a; H) = r N!

ar = a ∩ H!

r = | a : a ∩ H |!

r

o(an; H) = (r, n) n ZD

. o(a) ! o(a; H) | o(a)D

0 o(a; H) = ∞ ! o(a) = ∞ a ∩ H = 1D % H ≤ K ≤ G ! o(a; K) | o(a; H) o(a; H) D

& am H (m N)! m o(a; H)

I a, b G, o(a) = α, o(b) = β, o(a; b ) = α1, o(b; a ) = β1!

( αα1 = ββ1 D

. bβ1 = ar! α1 = (α, r)D 0 bβ1 = ar ab = ba!

αβ1 o(ab) = (α, β1 + r) .

! a, b G ) o(b), o(ab) |a, b|,

( o(a) = 12, b7 = a9, ab = baD

. o(a) = 20, b4 = a6, ab = ba

" m, n N

( ) : Zm × Zn

. Zm × Zn Zmn (m, n) = 1.