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Белоногов. Задачник по теории групп

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, δ . >

( G E 4 !

. δ # : G #

0 Inn(G), δ D(Inn(G)).

Z(G) Φ(G) E G

:

! A ≤ B ≤ G! A B E

G !

( A B!

. B/A G/A

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gϕg1 CG(N ) g G

ϕ $ G/CG(N )

# G = AB, A G, CB (A) = 1, ϕ Aut(G), ϕ $

A B > ϕ = 1

$ α E # # ,

# G! N G α $ ,

N G/N > : x = 1 N #! o(x)

o(α)

( CAut(G)(Inn(G)) = CAut(G)(G/Z(G)). , α

G " ! gαZ(G) = gZ(G)

g G 3 ! , H

, G !

$

. I Z(G) = 1! Z(Aut(G)) = 1

I G E ! Aut(G) E

I H E H G! G = H × C!

C ≤ G

" G! " " : , αKer(α) = 1! " # ,

G = A ×B = A ×C ) ϕ : G → G

" # : g G

g = ab! a A! b BD : b

b = a1c! a1 A! c C

gϕ = a1 ) ! ! α : g → (gϕ)1g G

!

( ϕ End(G)! Gϕ = Bϕ ≤ Z(G)!

. α Aut(G), Aα = A, Bα = C

A B E ! α E , A B! β E

, B A I αβ Aut(A)! B = Ker(β) Aα

! G E ! n E $ αn E

g → gn (g G) ( I G !

αn Aut(G) (o(g), n) = 1 g G.

. I n {−1, 2, 3} αn Aut(G)! G

" n Z 7 G n !

xnyn = (xy)n x, y G,

αn : g → gn (g G) : , G

( n nm# " m N.

. A n# " # π(n): " π(n) : π(n) E ! n! π(n) E

# I G E 4 ! α Aut(G) o(α) = 2!G = G1 · G1! G1 = {g G | gα = g}, G1 = {g G | gα = g1}

$ G E ! α Aut(G) o(α) = p E

I α # 4 :

G! p | |G|

G E A E Aut (G)

! ϕ : g → g|A|(g G) , G

3 (( %- : ! G E

(o(g), |A|) = 1 g G. ) # , ϕ1

|1 | !

A

θ:= |A| α A α.

( θ End (G)

. θα = αθ = θ α Aut (G)! θ2 = θ 0 G = Gθ × Kerθ

% Gθ = CG(A)

& Kerθ = G1−θ = {g1gθ | g G}. e fdggdhi

I α E # : , G! 1+α : g → g1gα (g G) : , G

G H E Ω, Ω > C(G) =

C(H)

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G , A/A1 B/B1 #! ϕ(agA1) =

ϕ(aA1)g a A g G! CG(A/A1) = CG(B/B1)

(

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[a, b] $ ! 4

: G G

G I A B E G!

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[A, B] " n ≥ 3 : x1, . . . , xn G "

[x1, . . . , xn] := [[x1, . . . , xn−1], xn]

< A ≤ Aut(G) H ≤ G

[H, A] := h1hα | h H, α A .

> < " : a, b, c "

H ( [a, b]1 = [b, a]!

. [a, b1] = [b, a]b1! 0 [a1, b] = [b, a]a1!

% [a, bc] = [a, b] [ab, c] = [a, c] [a, b]c!

& [ab, c] = [b, ca] [a, c] = [a, c]b [b, c]! 5 [a, b] = [b1, ab]!

- A H [[a, b1], c]b · [[b, c1], a]c · [[c, a1], b]a = 1

ϕ E , G! a, b E 4 : A, B E 4

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[A, B] Ker ϕ.

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0 7 Gϕ

 

G

 

Ker ϕ

G

% G E

 

 

 

 

 

 

 

= 4 A, B C E

G > $ " [[A, B], C], [[B, C], A], [[

4 $ #

I A B E G ! [A, B] Z(G)![A, B ] = [A , B] = 1

H ≤ G

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. H G [H, G] H

0 H G G/H G H

[a, G] = [G, a] G " : a G

! I G E ! |G : G | ≤ |G : [a, G]| ≤ |CG(a)|

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# # G!

G

# M E G a G I M Ma E

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[a, G]

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( [AB, G] = [A, G][B, G].

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I M G! MG E G

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G .

G = A, B ! A B E G > ( [A, B] G!

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0 G = AB[A, B]! % G = A B [A, B]

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# A Sn "

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. [(i j k), (i l j)] = (il)(jk)!

 

 

 

 

 

 

 

0 [(i j k), (i l m)] = [(ij), (il)] = (ilj)

 

 

 

 

$ 2 # A4 S4

 

> S

= (A

n

) = A

n

 

n

5

 

n

 

 

 

 

 

 

 

 

# : A5

 

3 "

 

GLn(F )! F E

'5 H

 

( [tij (α), tjk(β)] = tik(αβ) i, j, kD

 

 

 

 

 

 

 

αβ

 

 

 

 

. [tij (α), diag(β1, . . . , βn)] = tij (

i

 

− α).

 

βj

 

> F E n N

( SLn(F ) = SL(n, F )! |F | > 3 n > 2

. GLn(F ) = SLn(F )! |F | > 2 n > 2

Z(G)

2 # SL2(Z2) (= GL2(Z2) ) SL2(Z3)

(. .(! " !

" ! A5 2 #

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H

G E ! 4 : g1, . . . , gn > # : g G g = ϕ(g1, . . . , gn)! ϕ(g1, . . . , gn) E # $ : g1, . . . , gn " "

# : gi mi (i = 1, . . . , n) >

α1, . . . , αn E ! ( !

αimi = 1 H = α1, . . . , αn C· >

G H! , #

µ: g = ϕ(g1, . . . , gn) → ϕ(α1, . . . , αn),

, ! x = ϕ(g1, . . . , gn) y = ψ(g1, . . . , gn)

!

µ : xy = ϕ(g1, . . . , gn)ψ(g1, . . . , gn) → ϕ(α1, . . . , αn)ψ(α1, . . . , αn) = µ(x)µ(y)

> : gi " (! Kerϕ < G,

H ! (. . G Kerϕ. > ! G < G

! a, b G.

( [a, b] = [a, cb] = [da, b] " c CG(a) d CG(b).

. A [a, b] = [a, bc] = [ad, b] c d ?

" G E ! G : Z(G) 2

< G

 

> G : !

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|

| |

 

# G E

h E #

4 4 : > l

G h ≤ l ≤ h2

$ ( I N G N ∩G = 1! N Z(G) Z(G/N ) = Z(G)/N.

. A G

G !

G = X , X G, N G. I "

: X N, G N.

g G [G, g] Z(G)

( x → [x, g] (g G) , G [G, g]

CG(g) A ! G/CG(g) [G, g]

. 2 G/[G, g] CG(g)

G E ! A E 4

! g G > ϕ : a → [g, a] (a A)

, A [g, A] CA(g) A !

[g, A] = g1gA |A| = |CA(g)| · |[g, A]|.

I [a, b] : a! ( [an, b] = [a, b]n " $ n!

. : a #! : [a, b]

#! 4 [a, b] a

I [a, b] a b!

m

 

n

 

 

mn

 

 

 

m, n Z!

( [a , b

] = [a, b]

 

n(n

1)

 

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n

 

n n

[b, a]

 

 

 

 

n N

 

= a b

2

 

 

0 a, b = [a, b] !

 

 

 

 

 

 

I [a, b, b] Z( a, b )!

[a, bn] = [a, b]n[a, b, b](n2) "

n ≥ 2

 

 

 

 

 

 

 

 

 

 

 

! I x, y G!

 

x2 = 1!

[[x, y], x] = [y, x]2

" I G = G , Z(G/Z(G)) = 1.

# I N G!

 

[N, G ] = 1! CN (g) G " g G.

$ G = A B A1 ≤ A. I B $ A1 ,

[A, B] CA(A1).

I G = A B! G = (A ∩ G ) (B ∩ G )! B ∩ G = B

K E G ! K ∩ Z(G) = 1 > " # # N G!

# K! KN/N ∩ Z(G/N ) = 1

I a G a G! CG(a) G

I A B E G A/B E $ ! NG(aB) G " a A

I G !

[[a, b], c] [[b, c], a] [[c, a], b] a, b, c G.

I P Sylp(G)! G = G NG(P )

! I G E G = p! p E !

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G #D

. p E # # G! |gG| {1, p}

g G

" 7 ! # 4

" !

# G = AB! A B E G! a A!

A ∩ [a, B] = 1 > CG(a) = CA(a)CB(a)

$ I α E # ! ((j%(! $

# , G! α # G .

I A ≤ Aut(G), [G, A] E A

G.

I N G, α Aut(G) N CG(α), N CG([G, α]).

N G A ≤ Aut(G) I A $ N G/N !α $ G/Z(N ) [G, A] Z(N ) A E

I G E ! A E Aut(G)(|A|, |G|) = 1, G = CG(A) × [G, A]. e kQbb*hlQmb

4% 4 5 #

3 G

G0, G1, . . . , Gn G ! Gi ≤ Gi+1

i {0, . . . , n−

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1 = G0 ≤ G1 ≤ . . . ≤ Gn = G

(1)

 

 

G = Gn ≥ . . . ≥ G1 ≥ G0 = 1

(2)

! $" " Z n

( I := {0, . . . , n−1} ; ( H

! Gi Gi+1 i ID

! Gi G i ID

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