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Белоногов. Задачник по теории групп

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! $#

E 4 ) ! #

G E n E $ $

G(n) n G! Zn(G) n G

Ln(G) n G " "

(i N)H

G(0) = G, G(i+1) = (G(i)) D

Z0(G) = 1, Zi+1(G)/Zi(G) = Z(G/Zi(G))D L1(G) = G, Li+1(G) = [Li(G), G]

) !

G(0) ≥ G(1) ≥ . . . ; Z0(G) ≤ Z1(G) ≤ . . . ; L1(G) ≥ L2(G) ≥ . . . .

i

 

H(G) :=

=0 Zi(G) H Z(G)

G 7 G "

A B! " N

"! N A G/N B

A # G "

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GD

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Θ E : Aut(G) ; (

Θ Θ ! : Θ

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!

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pαqβ F # '.. pαqβ ! pq E !

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1 = G0 ≤ G1 ≤ . . . ≤ Gn = G.

( I H ≤ G!

1 = H0 ≤ H1 ≤ . . . ≤ Hn = H, Hi = Gi ∩ H,

H! 4 , Hi+1/Hi

, Gi+1/Gi (i = 0, . . . , n)

. I ϕ E , G!

1 = Gϕ0 ≤ Gϕ1 ≤ . . . ≤ Gϕn = Gϕ

# Gϕ! 4 Gϕi+1/Gϕi E , #

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> # =" !

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G E α E 4 #

, #!

( π(o(α)) ∩ π(G) = !

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! # , # #

: p p A

! " # E :

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. I N G N G/N ! G

0

G n

 

G(n)

= 1! G(n−1) = 1 A : G > G(1)

> . . . > G(n) = 1

 

 

 

G

 

# ( I G 1 = A0 ≤ A1 ≤ A2

. . . ! Ai+1/Ai Z(G/Ai) i = 1, 2 . . .

" " !

G! Ai Zi(G) i

. I G G = B1 ≥ B2 ≥ B3 ≥ . . .

! Bi/Bi+1 Z(G/Bi+1) i = 1, 2 . . .

" " !

G! Bi Li(G) i

0 I γ E , G! Zi(Gγ ) Zi(G)γ Li(Gγ ) Li(G)γ i N

$ N < G H ( G E n!

. Zn(G) = G! Zn−1(G) < G! 0 Ln+1(G) = 1! Ln(G) > 1

N G E

( , G

. I 1 < N G! [N, G] < N N ∩ Z(G) > 1

I A B E # #

G A < B! G!

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p " p

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7 S2, S3, A4, S4, Q8 D8 < #

G : G(i)! Zi(G) Li(G) (i N)! $ :

O

G , D2n D! ! G = a b !

o(b) = 2 bb = a1 ) G(i), Zi(G) Li(G)

i N A G O

 

a , o(a ) = 2n

 

 

 

Z2! ! A =

 

! G = A b

D(A)! A

 

( ) G(i), Zi(G), Li(G)

(i N) H(G)

 

n=1 n

n

 

 

 

 

 

. G/Zi(G) G

i N

 

 

 

0 G ! 4

 

% A E

G ! !

Φ(G) = A

 

 

 

 

 

" 7 An Sn n ≥ 5

# ! 5/!

$ p, q, r E G

" H

( |G| = pqb! b ND

. |G| = p2q2D 0 |G| = pqr

I |G| ≤ 200 |G| {60, 120, 168, 180}! G

2 (./ (6/

I a b E : G ! ab = bma ba = abn!

{m, n} Z! a, b 4 $

A # G

N ! G = N G

; ! "

4 : !

G E >

π !

( G = Gπ! Gπ G! 4 " 4 π: D

. G = Gτ ! τ E ! ! τ π

! ( A " # # 4 #

#

. A # # #

" V # # G

G 4

G

# G E ! G(n) = G(n+1)

n N > G/G(n) G(n) # #

N #! G/N

$ I h Z2(G)! ( h $ G !

. [hs, g] = [h, g]s " g G $ s

@ # G

$ G : #

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G # # 2

< G " H

( G

2!

. G Z(G)!

" a, b, c G!

0 [ab, c] = [a, c] · [b, c]

% [a, bc] = [a, b] · [a, c]

" a, b, c G

& [[a, b], c] = [a, [b, c]]

" a, b, c G

5 [a, b, c] = [a, c, b] " a, b, c G

a, b, c G m, n, r, s N I [a, b] = c

a b!

 

 

 

 

 

 

(

a, b

 

 

2

a, b

= c !

 

 

= an bn c

1

!

 

 

 

. (ab)n

2 n(n−1)

 

 

 

 

 

0 (ba)n = an bn c

21 n(n+1)!

 

 

 

 

% [am bn, ar bs] = [bn am, bs ar] = cms−nr

 

 

 

 

G E . a, b G

( =" 4 : G

. CG(a) G G/CG(a) [a, G]

0 [an, b] = [a, bn] = [a, b]n " n N

% I exp (G/Z(G)) ! exp (G )

I 1 = A0 < A1 < . . . < An = G E $#

# G! exp (Ai+1/Ai) exp (Ai/Ai−1) i = 1, . . . , n

! A # 2 " : a b : ab, a2b2, a3b3, . . .

" < ! # 3

# H ≤ G H

# # H > Φ(H)G Φ(G) A

! H G! Φ(H) Φ(G)

$ I G E G/G $ ! G $

A # G G ≤ Φ(G)

G E > Φ(G) F

! N G N/N ∩ Φ(G) ! N

A # # G , G/Z(G) # ! G

G !

4

# # G

( , #

. ; #

# # #

 

A # # p

" p

! G E

# # N =

P Q! P Q E !

4 P E

G > Q G!

G = P NG(Q)

 

 

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G

m

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# G E

 

( ; # S(G) G

G

. 9 F (G) G

G

$ > A # # G CG(F (G)) F (G)

G E α E ,

(G) #! (o(α), |G|) = 1 > α $ F (G)!α = 1

G E Z(G) < G > Z(G) < F (G)

I A B E ! (|A|, |B|) = 1!

H(A × B) = H(A) × H(B).

I G = G ! H(G) = Z(G)

G E > H(G)

N G ! Z(G/N ) = 1

K ≤ G i N

( K ∩ Zi(G) Zi(K) K ∩ H(G) H(K)

. I K/K ∩ Zi(G) ! K

0 I G K/K ∩ Zi(G) p! K p p E

! I K E G! qe ]

" G E $! N G N ≤ H(G) > H(G/N ) = H(G)/N

# 7$ # G

G

$ A # G $ H(G)

G

2 p P # G

H(G) ! $ " # p : G

|G : CG(P )| = pa, a ≥ 0

 

I G E ! π(H(G)) = π(Z(G))

 

G E

)

! ! H(G) :

p! Z(G) : p

 

 

 

G = AB! A B E G

( [A, B] = G

 

 

 

 

 

 

 

. [a, b a1b1 = [a, b]b1a1

" a, a

 

A b, b

1

B

0 G

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1

 

 

 

 

G E 2

r sgW

( I H G K G! H ∩ K G

. I H G K G! HK G

0 I G ! H G K G, H, K K

( #

#

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! G E M E 4

! M G

(0 55 ! d E # # k E

: M > d > 1

k < |G|! G F ! G

" " P # , # G/P ! E " " H, # G/H P Syl(G)

" G E H ≤ G I

G! H!

4 ! # 1 H . . . G!

, ! " !

# p1, . . . , pl E 7 !

G (p1, . . . , pl)! G

# 1 = G0 < G1 < . . . < Gl = G #! "

i{1, . . . , l} |Gi/Gi−1| pi

G A B

I A B " !

A B G

!$ A # "

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