Белоногов. Задачник по теории групп

AU/W : G → GL(U/W ) A!

AU/W (g) : u + W → uA(g) + W (g G, u U ) ..8 ;

#

# D ..5 $

# .. -

+ A G F

5 G F G GL(1, F ) GL(V )! V V (1, F )! g G

M E G → GLn(F ) *

M " ,$

Mij : g → M(g)ij (g G), i, j {1, . . . , n}.

J ,$# G F $ ϕ1 + ϕ2 : g → ϕ1(g) + ϕ2(g), f ϕ : g → f ϕ(g) ϕ! ϕ1! ϕ2 E ,$! f F

F F |G| ) (G → F ) CF (G → F ) E (G → F )!

& ! ,$#!

4 : G

# G F

! char (F ) |G|!

+ G "

G C V G C

& G 3 !

" χ! "

H deg (χ Ker (χ χ ,$ G C " G ;

G ! & G V

G C &

G 1G

Irr (G) E GD

H

αβ : g → α(g)β(g) (g G);

Irr(α) := {χ Irr(G) | (α, χ)G = 0} Irr(Ξ) := {Irr(ξ) | ξ Ξ} I

{C1, . . . , Ck} E :

G! gi Ci Irr(G) = {χ1, . . . , χk}! (k × k) $ ‚ ‚ij = χi(gi)

D G " "

Irr (G)! # " D! Φ

G E G!

# " Φ

I ‚ E $ G! Φ Irr (G) D |G! ‚(Φ, D) $ ‚! " !

" Φ! $! " :

D A ! D Φ # " ! $ ‚(Φ, D)

& ‚ G

! ! G

A

# 4 #

.. -- .. -6

M1 M2 E G F

; H

( M1 M2 E $

M1 ≈ M2D

. $ S F !

S−1M1(g)S = M2(g) g G.

A1 : G → GL (V1) A2 : G → GL (V2) E

G ; H

( A1 A2 E $

A1 ≈ A2D

. , σ V1 V2 #!

σ−1A1(g)σ = A2(g) g G.

A G F "

" : # A ≈ B A B

I A B E : G! ( Ker (A) = Ker (B)D

. A(G) B(G)

( G "

. V " G # , $#

A B E G # m n

F ) ,$"

A & B : g → A(g) & B(g) (g G).

" < # A B .. - χA B = χAχB

# A E G → GL (V )! U W E A(G)

V

M(g)s = sN(g) g G,

s = o $! m = n s

> I X E G

! $ X(G) E $#

X E G → GLn(F )!

F E > $ Mn(F )!

# $# X(G)! #

! F E

( A # F "

(

. I X E G F ! deg X = 1 Ker X G .

" S(M) (G → F ) ,$#G F ! 4 ,$

M! &

M

( ,$# :

# "

. I R E G

F ! S(R) = (G → F )

# # F # G E

M E G → GL(n, F )

F ; H ( M D

. ,$# Mij M

n2 : # D

0 F [M(G)] = M(n, F )! M(G) n2 #

F $

# χ(xg)χ(g) = |G|χ(x) x G χ Irr(G) g G χ(1) "

$ I X E $ G g1, . . . , gk E

4 : G!

CF (G) " # H

(α, β)G = (β, α)G, (α + β, γ)G = (α, γ)G + (β, γ)G,

. V χ G (χ, χ)G = 1 0 I χ ψ E G! Ker(χ + ψ) = Ker(χ) ∩

Ker(ψ)

χ E G

( (χ, χ)G ≤ χ(1)2

. (χ, χ)G = χ(1)2 O

(

: #

. I χ, λ Irr(G) λ(1) = 1! χλ Irr (G)

0 L λ(1) = 1