Белоногов. Задачник по теории групп

G(f ) := {g GL(V ) | f (xg, yg ) = f (x, y)}.

( G(f ) E GL(V ) f

. G(f ) Sp2n(F )

# 2 V = V (2, F )

# " , f (x, y) = x1y1 + x1y2 x = (x1, x2) y = (y1, y2) E

V

( + f (x, y) = 0 f (y, x) = 0

. G := {g GL(V ) | f (xg, yg) = f (x, y)}O

# ( U (n) := {g GLn(C) | g−1 = g¯}

GLn(C) (¯g)ij = gij # #

n

. SU (n) := {g U (n) | det(g) = 1} U (n) U (n)/SU (n) T ! T

E # ( (-

# α E , F a Mn(F )

G(a, α) := {g GLn(F ) | ga(gα) = a}

gα E $ (gα)ij = (gij)α

( G(a, α) E GLn(F )

. F = Fq2 > − : x → x¯ = xq (x F )

, F ! !

GUn(q) := {g GLn(Fq2) | g−1 = g¯ = G(en, −)

E GLn(Fq2) !

n Fq 4H GU (n, q2)

# ! . V = V (n, F )

τg,v : x → xg + v (x V )! g GL(V ) v V

( J AGLn(F ) := {τg,v | g GL(V ), v V }

SV V

. AGLn(F ) = T G! T V + G GL(V )

# " ( H; = {(aij ) GLn(F ) | a11 = 1, a12 = a13 = . . . = a1n = 0} > H ≤ GLn(F ) H AGLn−1(F )

. 2 # ! , " AGLn−1(F )! SLn(F )

# # 7 AGLn(F ) F = Fq " AGLn(q)

( AGL(1, 2) S2

. AGL(1, 3) S3

0 AGL(1, 4) A4

% AGL(2, 2) S4

#$ G = GLn(F ) g G

( I F ! g G

4 fio(g) = 1 i! o(g) D !

G = T g! T = Tn(F ) & .5

g G

. I F E " p! : a (

p: f1 = . . . = fn = 1

# F E ! g E :

m G = GLn(F ) char(F ) m

# I a E $ GLn(R)! det(a) =

±1

B 4 .

# ( I g E : GL2(Z)! o(g)

1 π7 5 π7 <

7 > :

A " : , G " ! p E

! π E !

" H π(H) #

# HD π E !

πD

Oπ(G) E π GD

Oπ,π (G) E # G Oπ (G/Oπ(G))D

Oπ(G) E G! ,

# π H ! 4 π : G

7 G

π ! π"!

# ! # , π

! π! π D

π ! π

π D

π ! G := G/Oπ(G) H CG(Oπ(G))

Oπ(G))

< π# G π

1 = P0(G) M0(G) P1(G) M1(G) . . . Pl(G) Ml(G) = G

" H Mi(G)/Pi(G) = Oπ (G/Pi(G)) (i = 0, 1, . . . , l)!

Pi(G)/Mi−1(G) = Oπ(G/Mi−1(G)) (i = 1, . . . , l) Z l π

G lπ(G)

A ! π = {p} : p π

H ≤ G 7 !

H * G! H NG(H) =

HD

H G! H, Hg g " g GD

H G! H Hg H, Hg " g GD

H T I G! H ∩ Hg = 1 g G \ NG(H) ) ! 9 # > 4

'(0 " # , H

π π# π

#

$ > @ G "

N

( A G N G = N H! H ≤ G

. I N G/N !

N G G

9 # > 4

N G/N

π G GD

π G GD

π G # π#

GD

π G # π

# G

$ N G π := π(N )

( =" π : G!

N ! : N

. Z 4 π : G

4 π : G/N

0 I x E π : G! CG/N (xN ) = CG(x)N/N

$ G E ! H ≤ G ϕ E ,

G #! (|H|, |Ker (ϕ)|) = 1 H Ker(ϕ)

>

( NGϕ (Hϕ) = NG(H)ϕ!

. CGϕ (Hϕ) = CG(H)ϕ

$ A B E G ! A, B = A B

(|A|, |B|) = 1 > CG(AB/A) = CG(B)A

$! > π π A !

p p

$" ; H ( G p!

. G/Op (G) p!

0 CG(S) Op ,p(G) # p# S

Op ,p(G)!

% CG(S) Op ,p(G) # p# S Op ,p(G)

$# α E " # H π!

π! π

( 3 α ,

, ! " # # α!

" # α

. 3 # π π = {p}

,

$ $ G = H A! (|H|, |A|) = 1

H! A > H = [H, A] CH (A) I ! ! H !

H = [H, A] × CH (A)

$ > A E π , π G

A G p π(G)

( G A" " p

. =" A p G

: CG(A)

0 A p G

# A# # p G

$ A E π , π G! 4 A

G P E A p

G > CP (A) E p CG(A)

$ G = H A! (|H|, |A|) = 1 HA

( I A # H # 4 A# ! A # :

:

. I A # 4 :

H! A # : :

$ I N G G/N ! G

A ! G = AN

$ I P Syl (G)! P Φ(G) ! ! π(G/Φ(G)) = π(G)

$ ) ( @ ./ ( !

"

I N G! G H "!

. " A1 B1 ! A = A1N ! B = B1N

(|A1|, |B1|) = 1

$" ( I G/Φ(G) π! G π

. I G/Φ(G) π! G π

$# N G G/N E π1 π! π1 π > π1 π K G ! G = KN

$ $ N G! N E π H E π π

N

( H G

. G = N NG(H)

$ I H G! G = HG NG(H)NG(H) G

$ < A G H ( A GD

. I A ≤ H ≤ G! NG(H) = H A

! 4 H

$ N G H ≤ G

( I N ≤ H! H H/N G/N H

G

. I H HN HN G! H

G

$ N G N E p I K/N E

G/N ! K = N K1! K1 E G

$ N G E

( G 4

#

. I C E G!

C G!

N G G/N ! G = N C

$ < # S4! GL2(3)! D2n D∞ # .

$! I H E # # G!G = NG(H)G

$" G E p "

p G " p ( > G p (

$# G E p ! #

p ≤ 1 > p G ≤ 1

$ $ I p G p!

4 p (

$ ( π

G Oπ,π (G)

. Z π

GO

$ ( I Oπ,π (G) ≤ H ≤ G! Oπ,π (G) ≤ Oπ,π (H)

. I H G! Oπ,π (H) ≤ Oπ,π (G)

$ P E p Op ,p(G) I

H := Op ,p(G)CG(P ) p! G p H = Op ,p(G)

$ CG(Op(G)) Op(G) G p Op (G) = 1! P Sylp(G)! A E P ! P

G AG ∩ P A > A G

$ P E p G P1 Sylp(P CG(P )) > NG(P1) E p F ! P2 E p

G! P1! NG(P2) p

CG(K) K

. π# G! #

CG(Oπ(G)) CG(Oπ (G)) Oπ(G) × Oπ (G)

$" I N G! N Φ(G) Oπ (G) = 1! Oπ (G/N ) = 1