Белоногов. Задачник по теории групп

. a, b am, bn E ! mn = 1

$ I ϕ E , G = X | R !

Gϕ = Xϕ | R(ϕ) O

W W(X) ψ E X

Xψ > Xψ W (ψ) E , # X W

. G , , X R

( Q8 a, b a−1ba = b−1, b−1ab = a−1 a, b a4 = 1, b2 = a2, bab−1 = a−1

. a, b a4 = b4 = 1, b−1ab = a−1

( D∞ a, b a2, b2 c, d c2, c−1dcd

. D2n a, b an = 1, b2 = 1, b−1ab = a−1

a, b a2, b5, a−1bab−2 O

p E G = a, b, c ap = bp = c2 = 1, ab = ba, ca = bc O

! Q+ a1, a2, . . . , an, . . . a22 = a1, a33 = a2, . . . , ann = an−1, . . .

" Q(p) a1, a2, . . . , an, . . . ap2 = a1, ap3 = a2, . . . , apn+1 = an, . . .

# Zp∞ a1, a2, . . . , an, . . . ap1 = 1, ap2 = a1, . . . , apn+1 = an, . . .

$ R R(X)! w W(X) y / X > X {y}

R {y = w} X R

R R(X) R x = u!

x X u W(X \ {x}) > X R X \ {x} R1 ! R1

R # R : x

u

X \ Y > G = X \ Y | R1 ! R1 E #!

" # R : y Y

wy

a, b, c b2, (bc)2 x, y, z y2, z2

a, b, c b−1cb = a, c−1ac = b, a−1ba = c a, b aba = bab

a, b, c, d ab = c, bc = d, cd = a, da = b ?

( S3 a, b a3 = 1, b2 = 1, (ab)2 = 1 x, y x2, y2, (xy)3r, f f 2, f rf r−2

. A4 a, b a3 = 1, b3 = 1, (ab)2 = 1 c, d c3 = 1, d2 = 1, (cd)3 = 1

0 S4 a, b a4 = 1, b3 = 1, (ab)2 = 1

% A5 a, b a5 = 1, b3 = 1, (ab)2 = 1

! A G = a, b | a−1ba = b2 : a b "

" G = a, b | am = 1, bn = 1, bab−1 = as ! {m, n, s, } N > o(b) = n, o(a) = (m, sn − 1) G = a b

# ) G = a, b a3 = b3 = 1, ba = a2b2

H = a, b a3 = b6 = 1, ba = a2b2

$ ( I a4 = b4 = 1 ba = a2b2! ab = (ba)2 (ba)5 = 1

. a, b a4 = b4 = 1, ba = a2b2 c, d c4 = d5 = 1, dc = cd2

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a, b a5 = b5 = 1, ba = a2b2 c, d c5 = d11 = 1, dc = cd3

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G 4 H ! H G

G HH

G = a, b | a4 = 1, a2 = (ab)2 = b2 ! H = a2 D

G = a, b, c | a2m = b2 = (ab)2 = c2 = 1, ac = ca, bc = cb !

H= am, c ! (m N)D

G = a, b | a−1ba = b−1 ! H = a2 D

G = a, b | a2 = b2 ! H = a2

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G = a, b, c, d | a2 = b2 = c3 = d3 = 1, a−1ca = d, ab =

cb > o(a) = o(b) = o(c) = 3 G = ( a × c ) b = ( b × c ) a

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{an | n Z} {an = a2n+1 | n Z} !

( G = an an < an+1 n ND

n Z

. ϕ : an+1 → a (n Z)

, ϕˆ GD

0 G ϕˆ Hol(G) an (n Z)

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! G = F ({x, y}) M = {xnyxn | n Z, n ≥ 0} >M = M 3 ! #

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p! m, n E

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cl(P ) E P

d(P ) E "

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m(P ) E P ! n !

P : " pn

rs(P ) E $# P ! max{m(H/K) | K H ≤ P }

Ωi(P ) := g P | gpi = 1 Ω1(P )

P

i(P ) = gpi | g P

I a E # P !

A(P ) := {A | A ≤ P, A = 1, |A| = a}!

J(P ) := A | A A(P ) E ? P

SCN (P ) E

P ! ! $

" $ P

P

SCNk(P ) := {A SCN (P ) | m(A) ≥ k} p P

n − 1

n ≥ 3! x = e2πi/pn y = e2πi/p ! Zpn = x Zp = y !

ϕ E # , Zp Aut (Zpn )

Mpn := x ϕ y ! ϕ(y) : x → x1+pn−2 (n ≥ 3) A ,

E(p) := a, b, c ap = bp = cp = 1, ac = ab, bc = b (p > 2)

2 ! !

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0 7 p p2

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( Φ(P ) = P 1(P )

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0 d(P ) = d F ! X P

" P X = {x1, . . . , xd}

P/Φ(P ) = x1 × . . . × xd ! xi := xiΦ(P ) P/Φ(P )

$ !

! P E . ( Φ(P ) = 1(P )

. I P = AB! A ≤ P B P ! Φ(P ) = Φ(A)Φ(B)[A, B]

! I N P ! ( d(P/N ) ≤ d(P )!

. d(P/N ) = d(P ) N ≤ Φ(P )

! N P ; H ( P \ N : ≤ pm! . Ωm(P ) N !

0 Ωm(P ) = Ωm(N )

! ! I P = x, y = x (x, y P )! x, xy < P

! " P E p3

( P E :$

. xP = xP " x P \ Z(P )

0 I p = 2! P , D8 Q8

% I p > 2 exp(P ) = p! P E(p) ! ! P = ( a p × z p) b p! ab = az, zb = z

& I p > 2 exp(P ) > p! P Mp3 5 M8 Q8

- < 0 K & !

! Ω1(P ) 1(P )

! # ( I |P | = pn! cl(P ) ≤ n − 1

. I cl(P ) = n − 1 P E !

|Z(P )| = p

0 I |P | = pm! cl(P ) ≤ m + 1

!$ I cl(P ) = c exp(P/P ) = pm! exp(P ) ≤ pmc

! ( I P ! d(P ) = m(P ) = rs(P )

. rs(P ) = max{d(H) | H ≤ P }

0 I N P !

d(P ) ≤ d(P/N ) + d(N )!

m(P ) ≤ m(P/N ) + m(N )!

rs(P ) ≤ rs(P/N ) + rs(N )

! I |P/P | = p3 P E $#! rs(P ) ≤ 3

! P = a, b ! o(a) = 8! b2 = a4 ab = a−1! P Q16

2 #

( Z(P )! P ! Φ(P )!

. : 4 : P ! 0 P !

% " ,

, P ! & cl(P )! m(P )! rs(P )

! P = a 4 b 4! b−1ab = a−1 ( P = a2

. Z(P ) = Φ(P ) = Ω1(P ) = a2 × b2

0 L P

% a2b2 E $

. P ! ! P

& P/ a2b2 Q8

! P E p4! 4 : a b

! o(a) = o(b) = p2! ab = a1+p ) ( Z(P ), P cl(P )!

. Φ(P )! 0 Ω1(P )

! 2 $ p4 p > 2

, Mp4 ! #

! ! > 7 pn! " $# #

p! , # H Zpn (n ≥ 1)! Zpn−1 × Zp (n ≥ 2)!

Mpn (n ≥ 3)! D2n (n ≥ 3)! Q2n (n ≥ 3)! SD2n (n ≥ 4)

Φ(P )!

$ !

Ωi(P ) (1 ≤ i ≤ n − 1)!

< ! apn−2 P

! # P Mpn ! n ≥ 3 pn = 8! ! P = a pn−1b p ! ab = a1+pn−2

( P = apn−2 Zp

. Φ(P ) = Z(P ) = ap Zpn−2 0 cl(P ) = 2

% P p + 1 H ap × b abj !

% A : P \ a " .

& . " 0 4 : M a2n−2 ! b! ab! 4

(a2n−2 )P = {a2n−2 }!

bP = b a2 ! CP (b) = Z(P ) × b ! (ab)P = ab a2 ! CP (b) = Z(P ) × ab

- J a "

P

6 =" 4 "$! Z(P )!

" P

(-./ A ! !

P " "$"!

P/Z(P ) D2n−1 !

cl(P ) = n − 1 P E !

P H

P \ a 4 : %! m(P ) = 1! rs(P ) = 2

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P

6 P \ a : . %

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P ! P . 4 "$# .

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(/ P ( (

6 I V E

P ! CP (V ) = Z(V ) |NP (V ) : V | = 2

(( I D E : 2m ≥ 8 P !

$ D a (. ) m(P ) rs(P )

! ) p! " $