Белоногов. Задачник по теории групп

( P = a ◦ b ◦ c ! ! |P | = p9D

. Z(P ) = Φ(P ) = ap × bp × cp D 0 P = ap × bp2 × cp3 D

% Aut(P ) E p

J A V #

" # > 7 "$ P E p! M P γ E

, P Z(M) M

( ) α : x → xxγ (x P ) , p

P

. I Z(M) Z(P )! α E # ,

0 I |P | > p! p | |Out(P )|

"$ + J $ 5j|QbMlc* "

H

( > J P Epn A E p

Aut(P ) > " # A# P1 P P

A P2 ! P = P1	× >	"
	P2
. G = P H! P Epn H E p
# G P1 P P		G
P2 ! P = P1 × P2

" G = K A E ! (|K|, |A|) = 1 K = L×M E ! 4 L A

> K = L × M1! M1 E A K	A <
J
" G E P E	:

pn G > G/CG(P ) ,

# Tn(Zp) '5 A ! p
G/CG(P ) , Zp−1 × . . . × Zp−1
		n
" I G = P H E	! P Syl2(G)
CH (P ) = 1! H

" G = E D! E Ep2 ! D D2q! p q E

q | p − 1 ! E D

p > G = AB! A E pq B Z2p

" G = Hol (E9)

( G = E H! E E9 H GL2(3)

. A G : 8

0 G : 0 % G : 5

" ) Aut(Z4 Z2)

" ! + Aut (D(Z4 × Z4) )

/ ; 9

! $ #

! '5

A " : , F E ! q E

! Fq E q : ! p E m, n N

V V (n, F ) E n

F

GL(V ) E # V

'5! " # # B V

C #C , iB GL(V ) GLn(F )!

" # $ B det(α) = det(iB (α))

B

SL(V ) := {α GL(V ) | det(α) = 1} GL(V )

Z := Z(GLn(F )) )

P GLn(F ) := GLn(F )/Z E !

n F !

P SLn(F ) := SLn(F )Z/Z ( SLn(F )/Z(SLn(F ))) E

n F "

P GL(V ) P SL(V )

) V V (n, F ) " !

n − 1 E V

A GLn(F ) " Tn(F )! U Tn(F ) Diagn(F )

! D '5

F = Fq GLn(q)! SLn(q)! P GLn(q)! P SLn(q)! Tn(q)! U Tn(q) Diagn(q) GLn(F ), SLn(F ), P GLn(F ), P SLn(F )

Tn(F )! U Tn(F ) Diagn(F ) + P SLn(q)

" Ln(q)

2	$! 4 '5H
en E n × n $D
o E $	D ,

n

tij (f ) := en + f e(i, j)! e(i, j) E n×n $! # (i, j)

$! D

di(f ) := en + (f − 1)e(i, i)

A AGLn(F ) (8 %-!

Sp2n(F ) (8 %( U (n)! GUn(q) (8 %&! (8 %5 A (8 ( , 5 (.

# > n ≥ 2 F E

( I V E n F !

GL(V ) GLn(F )

. GLn(F ) = SLn(F ) A! A F · 0 Z(GLn(F )) = {f In | f F ·} F ·

% Z(SLn(F )) = Z(GLn(F )) ∩ SLn(F ) & SLn(F ) = tij (α) | α F, i = j

5 GLn(F ) = tij (α), d(β) | α, β F, β = 0, i = j

- GLn(F ) = SLn(F )! |F | > 2 n > 2 6 SLn(F ) = SLn(F )! |F | > 3 n > 2

# G = GLn(F )! D = Diagn(F ) Monn(F ) E

$ G! $! #

$ # :

( D F ·

× . . . × F ·

n

F · Sn

.

NG(

D

Mon

F

D

H

! H

S

) =

n( ) =

n

# > ( |GLn(q)| = qn(n−1)/2(qn − 1)(qn−1 − 1) . . . (q − 1)! |Z(GLn(q))| = q − 1

. |SLn(q)| = |P GLn(q)| = |GLn(q))|/(q − 1)

0 |P SLn(q)| = |SLn(q)|/k! k = |Z(SLn(q))| = (n, q − 1)

# G = GLn(q)! S = SLn(q) P = U Tn(q)! q

p

( P p G S

. NG(P ) = Tn(q) = P D! D = Diagn(q) Zq−1

× . . . × Zq−1

0 NS (P ) = P D1! D1 =

Zq

1 . . .

Zq 1

×

×

n

−

−

−

% CG(P ) = Z(P )

n

1

×

Z(G)

# ( $ Mn(q) $! , " Fqn

. 7 GLn(q) $" qn −1 A

J GLn(q)

# V V (2, Fq)! q = pm! G = GL(V ), S = SL(V )!

v, w V \ {0} P (v) := {g G | vg = v}

( P (v) Sylp(G)

. P (v) = P (w) Fqv = Fqw

0 Sylp(G) = {P (v) | v V \ {0}}! |Sylp(G)| = q + 1 % NG(P (v)) = {g G | Fqvg = Fqv} T2(q)

& P (v) ∩ P (w) = 1! P (v) = P (w)

# ! V V (n, F ) L E

V

( < " x P SL(V ) σx : F v → F vx = F vg " g x 4

L

. 7 P SL(V ) # . L "

x → σx

0 I F = Fq! |L| = (qn − 1)/(q − 1)

# " g GL(V )! V V (n, F )

( I g " " V ! g Z(GL(V ))

. k {2, . . . , n − 1} g k

V $ gO

# # ) t GL(V )

V ! W V ! wt = w

w

W vt

−

v

W v

V t $

{0}

<

W +

< V +

( 3

B V ! # $

$ t tB = tn1(1)

. A $ GL(V ) #

0 n > 2 $ GL(V ) SL(V )

#$ > ( 7 P SLn(F ) ! P SL2(2)

P SL2(3)

. SLn(F ) ≤ GLn(F ) N G I |F | > 3 n > 2!

N≤ Z(GLn(F ))! N ≥ SLn(F )

# GL2(2) = SL2(2) P SL2(2) S3

# ( GL2(3) = SL(2, 3) a ! o(a) = 2

. SL2(3) = Q b ! Q Q8 o(b) = 3

0 Z := Z(GL2(3)) = Z(SL2(3)) Z2

% P SL2(3) A4 & P GL2(3) S4

5 . GL2(3)

O2(GL2(3)) = Q

- 3 . GL2(3) :

6 CGL2(3)(a) = a × Z E4 a E : # ( 8 CGL2(3)(b) = b × Z Z6 b E : # . (/ 3 4 "$# GL2(3) O

(( GL2(3) E . .

# ( GL2(4) = SL2(4) × a ! o(a) = 3

. SL2(4) A5

# 7 GL2(Z4) GL2(Z2 Z2) L

! ! $ , $

# ( GL2(5) = SL2(5) a ! o(a) = 4

. |Z(GL2(5))| = 4! |Z(SL2(5))| = 2

0 U T2(5) T2(5)O

% 3 . SL2(5) , Q8

& SL2(5) = AB! A SL2(3)! B Z5 A, a = A a 5 P SL2(5) A5

- P GL2(5) S5

# G = SL2(5)

( ) 4 : G <

# : g! # CG(g) |gG|

. ) 4 G

! # H! # NG(H)

0 L G

# ! ) $ " :

GL2(q)H

g1 =

1

0

a = 1!

0

a

g2 =

1

0

a = 0

a

1

# " G = GL(2, F )

( I g =

0

1

G!

a

b

CG(g) =

x

y

G

x, y

F

E .

ay

x + by

. @ G " 4 $ :

# # G = GL2(F )! F E

( ) "$ G!

. "$# G ! F = Fq!

#$ I char(F ) = 2!

( SL2(F ) "$"!

. GL2(F )

4 4

9 # > ! '(0

# G = SL2(q)

( |G| = q(q − 1)(q + 1)

. G : " q $

q − 1 q + 1

0 3 4 ! q!

G E $

# q = pm p = 2

( 7 SL2(q) " " .

. 7 P SL2(q) : " " . 0 7 GL2(q) ! , E8

# A # G {P GL2(pm), P SL2(pm)}

" " p P 4 NG(P ) )

p ! CG(P ) = P

# G = P SL2(7)! P Syl2(G)! Q Syl3(G) R Syl7(G)

( NG(Q) S3

. NG(R) E .(

0 G ! , " S4 % NG(T ) = T

& G . 4

5 G . 4 ! , S4 - ) G L

# > I q2 ≡ 1 (mod 5)! P SL2(q) !

, " P SL2(5)

# P SL2(9) A6

# ! ( |GL3(q)| = q3(q3 − 1)(q2 − 1)(q − 1)

. 2 # $ $ g = t31(1) GL3(q)

0 2 # Z(U T3(q))

# " G = SL3(3) τ = diag(1, −1, −1) "$

( CG(τ ) =

a

0

0

GL2(3)

0

b

c

a(be cd) = 1

−

0

d

e

0

. 3

SL3(3)

# # G = SL3(q)! q E 4 ! τ = diag(−1, −1, 1) E

"$ G

( CG(τ ) GL2(q)

. ϕ E # , G G/Z(G) > CGϕ (τ ϕ)

CG(τ )/Z! |Z| = (3, q − 1)

#$ G = GL3(q) H = a × b ! a = diag(α, α, 1) b = diag(1, β, β) {α, β} Fq \ {0, 1} 2 # CG(H) CG(H) ∩ SL3(q)

# G = GL3(q)! q = 2n! g = t31(1)

( CG(j) = U T3(q) D! D Zq−1 × Zq−1

. I g E g , G → G!

G := G/Z(G) = P GL3(q)! CG(g) CG(g)/CG(g) ∩ Z(G) CG(g) = Q a q−1! Q U T3(q) Z(Q) Eq

# G = SL4(q)! q = 2m! t = e2

o

(

t

E "$

e2

e2

G

. CG(t) = A B! A Eq4

B SL2(q)

# G = GLn(q)!

n = r + s!

{r, s} N

.

Ar, s :=

er

o

x E r

×

s $ Fq

x

es

Ar, s E

q

rs

>

:

G

# G = SLn(q)! q 4 !

t = diag(−1, . . . , −1, 1, . . . , 1) (k 4).

k

> CG(t) # N ! , #

SLk(q) × SLn−k(q)! " Zq−1

# 2 # $ C "$ tn1(1) G = SLn(2m)

! |O2(C)| = 2m(2n−3) Z(O2(C)) E2m

C/O2(C)O

# G = GLn(F )! F E .! t E

"$ G > $ f (t + en) + en!

f F ! G 4 O

# ! F E # .! G E

GLn(F )! $

0, 1, −1

( G = D S! D E Diagn(F )! , E2n !

S Sn

. ) D, a !

		0	1	0	. . .	0	0
		0	0	1 . . . 0 0
a =									(a		S)
					. . .
		0	0	0		0	1
		1	0	0	. . .	0	0

a # $ # #

# " K E $ $# < " #

$ a Mn(K) GLn(K)

G(a) := {g GLn(F ) | gag = a}.

$ K G(K, a) G(a) <

! G(a) E GLn(K)

# # ) A := G(a) B := G(b) GL2(F ) F

E ! a = 0

1 b =

0

1

M (8 06 ! !

1

0

1

1

( A = SL2(F ), char(F ) = 2!

A

D(F ·), char(F ) = 2 ;

. B

F +, char(F ) = 2 , B

F ·,

char(F ) = 2 .

#$ ) G(a) G(b) GL2(F )! a =

1 0

b = 0 1

0

0

0

0

# K E $ $#! n N a =

0

en

G(a) GL2n(K) M (8 06

−en

0

2n

K

Sp2n(K)

!

Sp2n(q)! K = Fq ( Sp2(2) S3