Белоногов. Задачник по теории групп

. ωχ E # , Z(CG) C!

! ωχ(z1z2) = ωχ(z1)ωχ(z2) " z1, z2 Z(CG)

0 ωχ(gG) E $ " g G % χ(1) | |G|

I χ Irr(G) χ(1) = 2! G

χ					Irr G
	χ(z)χ(g)				( )					χ(z)
( χ(zg) =						z Z(G) g G! 4					E
		χ(1)								χ(1)
o(z) $
. χ|Z(G) = χ(1)λ! λ Irr(Z(G))
0 χ(1)2 ≤ |G : Z(G)|
( Z(F G) E F G
:		˜	$ G			$		˜	, . . . , Ck E
. Z(F G) = F C1 . . .							F Ck! C1

0 3 " $ $ hijm !

i, j {1, . . . , k}	k
˜ ˜		˜
CiCj =	m=1	hijmCm.

% char (F ) = 0 I g Cm! hijm #

g = gigj ! gi Ci gj Cj

& F = C hijm $

GH

	|giG||gjG|
hijm =				χ(gi)χ(gj)χ(gm)		.
	G
			rr	χ(1)
	| |	χ I (G)

A # F G ! char(F ) | |G|! # : x #! x = 0 x2 = 0

χ E G H ≤ G I χ|H Irr(H)!

χIrr(G)

! χ E # G H ≤ G > H

χ|H

#

" H ≤ G! χ Irr(G) χ|H = χ(1)λ! λ E # #

H > [G, H] Ker(χ)

# H ≤ G ; H ( H Z(G)D

. < " χ Irr(G) λ Irr(H) #! χ|H =

χ(1)λ

$ χ ψ E G

( (χ ψ, χ ψ)G ≥ (χ, χ)G

. ( O

χ E G X E

G C M χ ) ,$" det χ

(det χ)(g) = det(X(g)) > det χ 4 #

# # G

χ E G g E "$ G

( χ(g) Z χ(g) ≡ χ(1) (mod 2)

. = χ(g) ≡ χ(1) (mod 4)! G "

.

G E ! # :

p ! E E : p

G χ Irr (G) >

( g G o(g) = p! χ(g) ≡ χ(1) (mod |E|)D

. χ = 1		χ(1)	≤	|E|	! G	G = p = 2
	G			2		| |

χ Irr(G)

( |χ(G)| = 1 χ χ = 1G

. |χ(G)| = 2 |G : Ker(χ)| = 2 χ(G) = {1, −1}

H ≤ G! n N ϕ(1) ≥ n ϕ Irr(H) \ {1H } > " χ Irr(G) χ(1) ≥ n! Ker(χ) H

χ E G ( I H ≤ G! χ(h) Z

h H

				χ(h)
. I H E B G! h H
Z
				1}!
0 I g G H = {gm | 1 ≤ m ≤ o(g), (m, o(g)) =
h H χ(h) Z
! I |G| 4 ! " # # #
χ G	χ =	χ	t „mahbd`*
" I G E 4 k E
4 : G! |G| ≡ k (mod 16) t „mahbd`*
# > χ E G a Aut (Q(ω))! ω E
# |G| $
( 9$	χa : g → χ(g)a (g G) E G

. χa Irr (G) χ Irr (G)

V χa χ

0 I x Q(ω) xa = x a Aut(Q(ω))! x Q

$ G! χ ω E ..&8

( Aut(Q(ω)) = {αn | n {1, . . . |G|}, (n, |G|) = 1}!

, αn ωαn = ωn A ! |Aut(Q(ω))| =

φ(|G|)

. χαn (g) = χ(gn) g G αn (

g G ; H ( χ(g) Q χ Irr(G)D

. g 4 G " : gm! m N (m, |G|) =

1D

0 g 4 G " : gm! m N (m, o(g)) =

1

A : $ # Sn E $

χ Irr(G) I χ E # Irr(G)

χ(1)! χ(G) Z

	( g G r E (x, y) : G !
g = [x, y] > r =		G	χ(g)
						E
	|	| χ I	χ(1)
		rr (G)
. I g E G! n N (n, o(g)) = 1! gn
G
	g G! X := {x G | x = g } χ E
G ! χ(g) = 0 >
			x		|,
	|χ(x)|2 ≥ 1			|χ(x)|2 ≥ |X
	x X		X
4 : E $
	G E ! χ Irr(G) l E :
g G ! χ(g) = 0
( l ≥ χ(1)2 − 1

. l = 0 χ(1) = 1

! > F # I g G |gG| = pa! p E

a N! G

" paqb! p, q E

a, b N! pαqβ N

# > N G

Irr (G | N ) := {χ Irr (G) | Ker χ N }.

( 3 χ → χ Irr (G |

N ) Irr (G/N ) ! χ (gN ) = χ(g) = χ(gn) = χ(ng)
g G n N
. N = χ Irr (G|N ) Ker χ
!$ N G g G		Φ := Irr (G) \ Irr (G | N )
( |CG(g)| − |CG/N (gN )| =
	χ Φ |χ(g)|2
. I CN (g) = 1! |CG(g)| = |CG/N (gN )| χ(g) = 0
χ Φ

! G = A × B < ,$# α β A B

α × β " ,$" G C

"! (α × β)(ab) = α(a)β(b) a A b B

( Irr (G) = {α × β | α Irr (A), β Irr (B)}

. I α β E A B ! α×β

E # G
! A = a1 × . . . × an E
( J Irr (A) $#	,$#

) ˆ

A &

A

. i E # # o(ai)

$ C : a = as11 . . . asnn A

,$" χa : A → C χa(at11 . . . atnn ) = s11t1 . . . snntn (si, ti Z) >

Irr(A) = {χa | a A}.

ˆ 0 ) a → χa (a A) , A
A
	% I	B ≤ A	!	α →	ˆ	,
ˆ	ˆ				α |B (α A)
A	B

| { ˆ | }

B := Irr (A B) (= α A α(B) = 1 ).

& ) B → B (B ≤ A)

A

ˆ

A

! $ Z4! E4! A4 S4

! P : G → SΩ E G

Ω + P

,$

χP : g → |{i Ω | iP(g) = i}| (g G).

( χP E G

. (χP, 1G)G = m! m E P(G)

! G = {g1, . . . , gn} E n ;

	: g		11 . . .	1n		(g G)
P →			g	g
		g− g1g . . . g− gng					χ I
								(G)
G G .. -% χP =								aχχ!
aχ N {0} A aχ							rr
					χ #

, # $ G

! H ≤ G! ·{Hx1, . . . , Hxn} E

G H ·{H1, . . . , Hm} E ! 4

H G >

P : g → Hx1 . . . Hxn

Hx1g . . . Hxng

(g G),

: g

1

1

1

m

(g G)

H

. . . H

Q → g− H1g . . . g− Hmg

"

G ! 4

χ (g) =

|gG ∩ H||CG(g)|

χ (g) =

|gG ∩ NG(H)||CG(g)|

.

P

|H|

Q

|NG(H)|

!! > H

≤ G T = (t1, . . . , tm) E G

H!

: G! G =

. .

. . .

Htm <

"!

A : H

→ GLn(F )

Ht1

"

,$"

G

G

2

AT : G → GLnm(F )

AT (g)

m

n (i, j)

A

(t gt−1)!

(x) =

(x)

i

j

A

A

x H A (x) = On×n x G \ H

( AGT 4 G F

7 ! AGT A T !

! AGT E G I T

! C T C "

AG AGT

. AG ≈ AG R G H T R " #

!" > H ≤ G T = (t1, . . . , tm) E G H < " # # ,$ ξ CF (H → F ) ,$"

ξG : g → m ξ (tigti−1),

i

ξ (x) = ξ(x) x

=1

H ξ (x) = 0 x

G

\

H

( ξG

,$ G F

T

7 ! ,$

ξG ,$# ξ! ! ξG

E G

. ξG(g) = 0! gG ∩ H =

0 I H ≤GK ≤ G! (ξK )G

= ξG

F ξ1, . . . , ξn

% i=1 fiξi

=

=1 fiξiG

" f1, . . . , fn

n

n

i

CF (H → F )

& I char (F ) |H|!

ξG(g) =

1

x

ξ (xgx−1) =

|CG(g)|

ξ(x).

|

|

| |

x g H

G

∩

H

G

H

!# ( I A E H G! χAGT = (χA)G

. I ξ E H G! ξG E

G & G

"$ H ≤ G! T E G H A E

H F

( # A AGT O

. I A E H F ! AGT E

G F

0 I A ≈ B $ C! AGT ≈ BTG $ CTG

% I ξ CF (H → F ) ψ CF (G → F )! ξGψ = (ξψ |H )G

" H ≤ G ρH E # H > (ρH )G = ρG E # G A ! 1G{1} = ρG

" G 9 H ≤ G! ξ CF (H) ψ CF (G) >

(ξG, ψ)G = (ξ, ψ |H )H .

" G E H ≤ G > #

# H H

G

" A ≤ B ≤ G > (1A)G − (1B )G E G

" H ≤ G ξ E ,$ G C ξ(g) =

|{(h, x) H × G | h = gx}| > ξ = |H| 1GH

" > T E T I # G H = NG(T ) > G " N "!

G = HN H ∩ N = H \ T .

: N = G \ (H \ H \ T )G

"! + ..65 " " 2 H ≤ G H∩Hg = 1 g G\H > G "

N "! G = H N F ! N = G \ (H \ 1)G

"" H0 H ≤ G

H ∩ Hg H0 H0g g G \ H.

> G " N "! G = HNH ∩ N = H0 : N = G \ (H \ H0)G

"# G = 1! g G χ Irr (G) > gG {χ} χ(g) = 0.

#$ D |G Φ Irr (G)

( I Φ E D G! Irr (Φ |0D) = |Φ| = 1!

Irr (Φ |0D) = Φ

. I Φ |0D C[Φ] + Cψ! ψ Irr (G)! D Φ!

D Φ {ψ}

# > G E D |G Φ Irr (G)

D− := G \ D Φ− := Irr(G) \ Φ 3 " H

( D ΦD

D

. d D

ϕ(d)χ(d) = 0

(ϕ, χ) Φ × Φ−D

0

ϕ(d)ϕ(g) = 0

(d, g) D × D−

ϕ Φ

1

d

%

ϕ0(d)ϕ(d)ϕ(d0) = ϕ0(d0) (ϕ0, d0) Φ × D

|

G

|

D

ϕ Φ

# G! D Φ E # ;

H

( D ΦD

. D− ΦD

0 D Φ−D

% D−1 := {d−1 | d D} ΦD

& D

Φ

:= {ϕ¯ | ϕ Φ}

# D! D1! D2 E G! Φ! Φ1!

Φ2 E Irr (G)

( I D1 Φ D2

Φ! D1 ∩ D2 Φ! D1 D2 Φ

D1 \ D2 Φ

. I D Φ1 D Φ2! D Φ1 ∩ Φ2! D Φ1 Φ2

D Φ1 \ Φ2

# A = X(Φ, D) E # , G Φ

Irr (G)! D |G)

( 1G Φ A (1, . . . , 1)

. 1 D A # $! #

$ A $! # #

Φ

0 I A " # $!

# , G

# D Φ

		1	1		1
X(Φ, D) =		4	0		1	.
		3		1	0
			−
		3		1	0
			−

( ! 1 D # $ :

D G

. < ! G A5

# ‚ E $ G! D |G Φ Irr (G)D D− := G\D Φ− := Irr(G)\Φ ‚(Φ, D) t $!

" : d1, . . . , dt !

T = diag (|CG(d1)|, . . . , |CG(dt)|)−1.

; H

( D Φ X(Φ, D) = tD . X(Φ, D) X(Φ, D) = T −1D

0 X(Φ−, D) = O

#! # ;

H

( D Φ X(Φ, D) = |Φ|D

. X(Φ, D)T X(Φ, D) = ED 0 X(Φ, D−) = O

#" I D Φ $ ‚(Φ, D) E

(D |G, Φ Irr (G))! D = G Φ = Irr (G)

## ; H ( G # ,

	1 . . . 1			s × t;
	1 . . .	1