Lectures on Character Theory 0914098179, 9780914098171

Citation preview

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Lectures on

Character Theory David

M. Goldschmidt

Department of Mathematics University of California, Berkeley

COPYRIGHT ©

DAVID

All rights reserved,

M. GOLDSCHMIDT

1980

ISBN 0-914098-17-9 Library of Congress Catalog Card Number: 80-81648

PUBLISH

OR PERISH, INC.

2000 CENTER STREET, SUITE 1404 BERKELEY, CA 94704 (U.S.A.) In Japan distributed exclusively by KINOKUNIYA BOOK-STORE TOKYO, JAPAN

CO., LTD.

iii

Table

Introduction Some

algebras

Induced

Some

fields

Blocks

.......

the

Schur

index

algebra

numbers

groups

Brauer

.......

Blocks

of on

oe

defect

Bibliography

Notation

.

.....

decomposition

defect

.

ew

correspondence

Generalized

Index

..

........

Defect

More

.

characters

and

commutative

Decomposition

The

and

algebra

characters

Splitting

Contents

.........+6-.

non-commutative

Group

of

2

numbers

one. groups

.....

........+2.e.e. «©

Index

©

©

«

«

«©

©

©

©

©

©

«©

........e..

af)

Introduction

It these

is

notes

term

important

are

"modular

ter

of

a

p.

In

fact,

character"

there

for

is

developed

by

not In

the

well

recent

is

has

been

[11]

and

Dade

[4]

in

play

a

remain

in

the

of

safe

years,

work

more

a

which

of

the

values

of

the

complex

groups

and

associated

lectures sity

Isaacs

of

genesis

which Chicago

and

I

of

Leonard

deep

and of

approach

Green

their this

notes

original

studying

1969.

Aliso

Scott,

and

in

the

papers.

We

characters

of

questions.

was

Brauer

a at

attendance

many

of

p

however,

on

manuscript by

Thompson

congruences

arithmetical

given

[6],

to

characteristic

irreducible

this

attended in

and

achieve-

many

much

These

Brauer's in

new

in

role.

interested

the

of

groups

that

by

modules

primarily

Indeed,

say

developed

are

finite

to

title

characters"

series

finite

such

pioneered

uncovered

powerful

important

spirit

was

on

better

group

remarkable

he

charac-

characteristic

much

which

the

understood.

Brauer's

much

of

a

the

information

properties

that

speaking,

of

A

subject

outset

to

field

little

properties

yet

a

decades,

It

the

refer

here.

In

five

subgroups. is

found

Brauer.

local-arithmetic

work

be

really

spanning

local

very

"Arithmetic

this

ments

to

over

is

at

Strictly

should

representation

be:

realize

mis-labeled.

representations would

to

the

series

of

the

Univer-

were

Marty

ideas

in

the

present

treatment

originated

tions

and

post-mortems

notes

are

almost

only

an

is

first

chapters

used

in

prisingly the

little)

of

the

lack

of

proper

to

most

of

of

defect

necessary

number

theory

chapters

results

I

for

the

energy

also

apologize

exercises final

cations good

For

and

chapter

to

a

can

good

a complete

groups

substitute.

10

to

Brauer's one.

(surdevelop

in

one

for

out

a lack

absence

manuscript

of

to

contributed them

exactly see

of

Of

form or another,

have

apologize

find

the

references.

bibliography,

2-rank

in

authors

only

intended

of

are,

other

to readers

was

flaws

bibliographic

and

where.

through

4

theory. serious

Significantly,

what

6

of

Chapter

blocks

many

having

review

of

but

not

a

give

Brauer,

The

presuppose

to

many

the

and

11

block

One

course

due

of

time.

that

theory.

Chapter

treatment

and

fundamentals

the

only

the

at

comprise

subsequently

5 presents

conversa-

background.

character

theoretic

various

self-contained

algebra

four

had

ordinary

Chapter

a

we

necessary

Character

is

entirely

undergraduate

The

the

which

in

here

who

[5].

did

I

must

interesting

applications.

on

"basic

1,

but

sets"

and

fortunately,

A appli-

[2]

is

§1

Some

Non-Commutative

In of

this

the

reader,

throughout

a

R

ring

scalar

Unless

A

in

A

A

has

Then

the

map

the

center

A.

of

.

required

a map

R-algebra

RxA

+A _

together

with

ring

A

toa

left

otherwise, that

is

will

R

with ring

these

integers

is

its is

unital).

of

an

R_

into

isomorphism,

image

in

the

automatically

notes,

and

assume

are

a homomorphism map

multi-

R-module.

A-modules

this

every

we

Q

2

field

center a

denotes

the

A

called

which,

When

that

An

ring

identify

rational

Z-

the

of

numbers.)

For

the

algebra.

rest

An of

(In

unital

the

matically

A

this

and case,

is R_

section, an

abelian

which the

let

A

group

commute

R-module

and

be

an

R-

admitting associate.

structure

is

auto-

determined

by

the

A-module

structure.)

An

A-module

is

one

which

is

non-zero

and

has

the

Jacobson of

the

trreductble

proper

of

A-module

actions

radical

is

with

(and

(Throughout

rational

which

commutes

@:1

A

Note

convenience

which

stated

a@-*

sometimes

the

ring.

with

converts

unit

of

algebra.

no

and ,

a

for

algebra

commutative

explicitly

that

ring

a

multiplication

addition

of

basic

together

in

review,

notes.

be A

plication

we

we

some

these

Let is

section

Algebra

submodules. of

annihilators

A

, of

JCA) all

We , to

define be

irreducible

the

intersection

right

A-modules.

Evidently,

(1.1)

J(A)

is

(Nakayama's

generated

a

2-sided

Lemma)

ideal

of

Suppose

A-module

and

M

A

M

itis

a

= MJ(A)

finitely

.

Then

M=0.

Proof: ated

We

non-zero

is

obviously

is

a

free

first

A-module

of

F

of

is

assertion

is

easy

consequence on

where

assertion

Let

the U

7

F

the

F,

along

mal

submodule

N

Py

+

F

is

a

let

N

of

Let

N

uN

F,

» then

(N4U)

get

F/NtU

= FL/N

module

of

Now

maximal

» then

a maximal

F if

be

fF,

F,

UN

9 Po

N

Fy

#

, then

rank

Fy

in

@

there

PF,

because

of

FP,

F

is

wC(U)

.

a

M

, and

maxi-

Then

containing

is

U.

U#F.

containing

Since

we

Fo

onto

F a

=

UtF,

>

maximal

sub-

U.

non-zero,

by

and

of

F

one

Fo

Fy

of

N+U

proper

proceed

=

projection

= N.

» Whence

of

hold

containing

submodule

were

submodule

#

we F

F

submodule. of

lemma,

This if

every

maximal

writing

the

that

module

to

gener-

submodule.

then

a

free

assumed

containing M

in

submodule

m(U)

be

rank,

,

finitely

assertion

Zorn's F

m(U)

maximal

Fy

of

wm

If

e

the

the

of

is

If

we

=

2

for

rank

and

to

every

maximal

contained

this

induction

a

finite

Since an

that

has

equivalent

A-module

submodule

show

could

get

choose

MJ(A)

a

CN

because we

M/N

must

is

have

The

of

(1.2)

M=

next

cations

A

Let

the

I(m)

is

a maximal m.

For

an

It

module,

for must

all be

of

M

Lemma)

then

=

MJ(A)

,

useful

appli-

and

M

ts

then

End, ¢M)

Since

invertible

=

zis

finite

maxi-

if M

A-module we

let

=A/I(m)~

= J

is

so

indepen-

0.

O

denote

the

M

ts

ts

so

End, (M)

of

R

an

an an

trreductble

R-divtston

algebra.

algebraically

dimensional

A-

over

closed R,

=R.

ker(€f)

End, (M)

right

I(m)

let

M

unique

,then

MJ

we

If

tf

a

JCA).

thus

End,(M)

fteld

=

that

,

with

m

and

follows

partieular,

f€&

some

M = mA _

A-endomorphisms

(Sehur's

Proof:

to

irreducible

Then

ideal

A-module

J

an

annihilator

of

In

be

{0}.

be

of

Then

M

I(m)

(1.3)

leads

commutative

J.

m©M-

R-algebra

M

0

ts

ideal

Proof:

dent

since

0.

observation

Suppose

let

But

(1.1).

mat

and

irreducible.

and

> non-zero

because

M

im(f)

elements is

are

A-submodules

of

irreducible.

End, (M) Moreover,

if

M

is

End, (M)

finite

, and

finite

dimensional

there

are

dimensional

braically

(1.4)

closed

division

of

A.

natural

algebras

tdeals

A

the

product

fg

in

End,(I)

gg".

So

ap:

I+

I

left

the

elements

fe

End,(1)

.

is

so

is

alge-

.

For by

Writing

map

a

all

,

,

E€

tis

then

"first

a

right

the

that

f£

,

then

given

by

right

>

is

the

ap

let

I

functions

no

tsomorphism.

Clearly, u,x

has

convention

uel, uu.

a€A

I

an

means then

A

that

ts

the

,

multiplication

choose

an

that

notational

a€EA

A-+End,(I)

Now

over

End,(I)

+ End,(I)

adopt

by

and

B=

We

val map

,

non-trivial

Suppose

Let

map

multiplication

no

R

O

Proof:

if

field

field.

two-sided

ideal

a

certainly

(Wedderburn-Rieffel)

proper

be

over

,

on

u,: u,

natu-

I+I1 ©

B

and

the

right,

we

have

(x) (audpf

Let we

v= find

(u)f that

.

=

(xa)upf

Since

(xa),

=

(u) (xa), f

€

B

as

observed

above,

(Cu) (xa)

f

=

Cuf) (xa),

and

hence,

Cau),f

of

AI

End, (TI)

in

ideal

of

1.

AI

We

onto.

Its

the

the

ensional

call are

that

A

This

under

the

Since

A

thus

that is

R

R

.

of

is By

for

and is

A

section,

and

=

A we

a right

contains

>

End, (T)

is

we

specialize

is

finite

will

and Since

the

.

We

if

there

irreducible

it

denote

dim-

mean

simple

coincide, We

0

A-module"

0

ideals.

semi-simple.

image

2-sided

AI

map

right

A/J(A)

is

proper

of

the

hypothesis.

field

J(A)

that

map

no

"A-module",

if

two-sided A

by

unital

semt-simple

shows

image

this

a

(x) (av),

natural

natural

trivial

=

has

the

the

R-dimensional

A/J(A)

by

and

that

proper

modules

-

remainder

over

A no

(av),

.

kernel

case

"finite

A

conclude

For

to

=

(xa)v

=

End,(I)

ideals,

=

is the

hr

obvious center

of

ZA).

(1.5)

JCA)

ts

tdealt

of

Proof: J(A)"

integer

is

the A.

Since a

n.

if

maximal

Moreover,

A

finitely

So

unique

is

J(A)"

J(Z(A))

finite

generated

#0

nilpotent =

J(A)

dimensional

A-module

for

some

right

for

n

NZCA) .

over any

R

,

positive,

, then

Lem

gcayn*? ¢ gay” nilpotent.

M for

If

any

we

get

right

(1.6)

MI

=

0. of

we

.

is

a nilpotent

each

N

is

Z(A)

obviously it

is

a

we

a

submodule

all

nilpotent

Cc

JCZCA))

nilpotent

easy

to

right

see

that

thus

and

A

of

ideal

right

A

Every

semi-simple.

of

MCN

patr

Then

there

A-modutes, that

sueh

LCN

, the

its

A-module

trre-

of

sum

direct

A-modules,

an

right

tdeal

irreducible

Then

homomorphism

I=

xA

given

tsomorphte

is

A-module

trreductble

a minimal

be

ts

a submodule

ductble

A-module

induction

0

that

Every

{0}.

n

MI

.

N=M@L

I

=

irreducible,

contains

J(A)

However,

extsts

to

get

NZ(A)

For

e)

JCA)

M

and

A.

above,

is

b)

and

Thus

Z(A)

a)

x€G€I-

obvious

since

Suppose

Let

an

cM

C JCA)

J(Z(A))

by

is

J(A)

J(Z(A))A

», then

M

the

of

mfr

If

MI

n

is

ideal,

=

M

that

such

JCA)

right

M=0.

ideals

ideal

nilpotent

whence that

because

any

integer

MI

By

is

positive

argument, conclude

I

module

any

is

by (1.1), and therefore

of

A

A-module,

so by

if ¢€a)

6: =

and

let

A+

I is

xa

we

the

find

that In

I =A/J

where

particular,

Now

if

can

be

I

NJ

Hence

A

0

We

can I

such

that

Jo

I

®

I,

By

the

and

I

by

argue

and such

I,

@

I

of

previous

since

I,

Hence,

we

A=tlI

®J,

is

ideal

J

A

A =

we

>» then

I* 2

A

#0

of

J

x

by and

ideal I,

CI

0

for

I,

=

Jo

WI,

Jy

~

=

I,

some

I,

right

there I,

is

SJ,

A

then

,

Put

and

of

bea

Jog

dim(I)

right

J

@

J 171,

get

any

I,

@

A

for

right

0

paragraph,

CJ,

A

that

on

I,

in

2

of

of

let

=

induction

I

ideal

INI

=

1,1,

6

already

shown

isomorphic .

by

some

c)

follows.

of

a).

Since

we

(necessarily

WJ,

15,)

that

1

some

Jo

NJ,)

every

for

now

know

that

J

right

ideal,

is

an

obvious

145,)

irreducible

A/J

b)

some

(I,

to

minimal)

Assertion

=

maximal is

1

irreducible.

for

(J,

=

2

have

have

module

a

Put

so

xa=0}

I.

Namely,

above.

1

right

because

exists

that

€ A:

maximality

then

by

fa

inductively

there

the

=

ideal

£J

I+J

ideal,

by

=

We

=

@

right

A=I@J.

Jy I

maximal

that

A

I,

exists then

now

right

ideal

a

minimality

of

minimal

=

by

is

= ker

minimal

so

=

ideal

a

chosen

(1.5). I

is

J

J

.

A-

right

complemented

assertion

consequence

Let

To

prove

Np

CN

a), be

we

argue

a maximal

By

induction,

Ny

= M

so

it

to

show

N is

.

suffices

Since

N/Ng

isomorphic result

Namely,

there

I

of

No

.

A

a

is

and

an

the

map

ww:

A+N@®dJ_

B=

imw

, then

such

primitive potents.

that

No

is

by

if

it

a

is

Ly

complemented

already A

,

I

J

with

,

then

.

of

1

rank

,

wl(o@@il)=1 verify

in

know

e”

is

it

the

ideal kernel

we

have

there

is

a

A

that

O of

=

>

nonsense.

right

@OT=A

to

so

general

N->

some

that

easy

submodule

minimal $9:

module

M.

of

of

dim(N)

.

ef it

c),

for

such

that if

summand

on

containing

we

consequence

free

idempotent

orthogonal

some

N@J+I

N=N,@®(BON)

e © A

for

A=1@6dJS

is

An

Lo

epimorphism

¢@1:

A

submodule

direct a

induction

irreducible,

exists,

Write

Since

@

is

to

desired

by

A

is

=e. fe

not

=

a

non-zero

Two 0

the

idempotents

, and sum

element

of

an

e,f

idempotent

two

orthogonal

are ec

is idem-

(1.7)

Suppose

that

A

algebratcally chosen

so

a)

the

for

Then some

conditions

complete

sided a

semt-stmple

closed.

that,

following

ts

ideals

complete

mintmal

integer

are

dtstinet

of

A

ts

set

of

patrwise

ideals

the

I.3

(1< SJs fj

the

tdempotents

is]

pairwise

.

and

>

C1

fs:

so

chosen

tirreductble

22g}

of

tndependent.

linearly

primitive

distinet

w;

tr

{05 Moree

ts

Then

representattons

Fi ofoo--- of,

be

that

such

closed.

number

finite

trreductble

can

means

(M, »Mo)

algebratcally

are

repre-

aé€EA

all

a

A-modules

two

which

Homp

©&

T

We

A.

afforded

stmtlar

funettons

trace

that

the

iff

define

representation

the

check

to

them

Notatton of

of

only

1

of

Hint:

k,

(2-1),

the

G;

X1 (8X5 (85)

Then character

of

of

Gy

G,

for

(2.2).

G,

G,

to

define

a

function

to

itself

classes

1


a

we

class

denote

on

2,8

defined by (2.11).

(3.1)

on

on

TfLhen

G.

(0° Wg

In

tf

particular,

then

9°

its

a

(0 Via

=

6

bea

elass

of

class

function

,

charaeter

tsa

character

a

p

and

HCG,

funetion

9%

Let

rectproctty)

(Frobentus

G.

on

formula:

(058) 55

functions

class

of

G

¥:

-

the

actually

is

function

by

product

6(xgx

9°

that

verify

to

F

be

the

by

G>

6

function

class

induced

F

H.

off

vanishes G

function

class G+

6:

Let

G.

numbers.

algebraic

of

field

F-

an

mean

to

continue

we

complex-valued

of

H

subgroup

some

of

character

of

subgroups

proper

of

those

and

G

of

characters

between

relationship

the

study

we

section

this

of

H,

Proof: direct

The

reciprocity

calculation.

We

formula

follows

by

have

(0°W6 = TET aan ¢g) Wey g Put X

h ,»

so

=

xgx

» then

does

h

G (O°),

=

mG

Since

is

summand

is

,

as

so

we

1 THT

1

|

|

x g

ranges

over

G_

for

fixed

get

z

|S]

constant

x&G on

independent

h&G

G

of

‘ @6C¢Ch)

Z

conjugacy

x

, and

-1

w(x

“hx)

.

classes,

vanishes

the

off

H

Hence,

(05,0), = Tar

=

IS|

Now

let

of

G

8

, it

particular, all eG

X © is

bea

is

ar

yee

IFl

character

clear

.

oth) PTRY

of

that

(8 Xiu

Irr(G)

Z

ney

Xu

is

H.

is

= (Osvya -

If

a

X

character

a non-negative

But

then G

a

character

of

The

following

formula

so

, by

for

is

is

a

character

of

H.

integer

ce® x)

> and

for hence

(2.12).

induced

In

Oo

characters

is

sometimes

(3.2)

useful.

Let

6

bea

elass

{X] 2Xpoe++ Xp}

G O°(g)

Proof:

because form is

hx;

constant

x

©

G

on

fhen

1=1

and

let

representagEG,

for

. -1 O(x,gx, 7)

2

can h

some

for

coset

t

=_

6(xgx7>)

IH] yee each

a set

of

HCG,

.

have

2

=

a°(g)

We

on

G.

in

H

for

tives

be

function

H

[H| new i=l

be

uniquely

conjugacy

vet

written

in

i.

Since

classes,

the

result

groups

and

6

show

that

and

©H

ext th +)

6(hx

>

-t =»:

some

the 60

follows.

(3.3)

HOCK

Suppose elass

funetton

CG on

are H.

ts

a

Then

ceX)& = 9S Proof:

CCON)E XD, (3.1):

By

(2.12)

= (0°X),

it

suffices

for all

to

X G Irr(G)

.

We use

SR

a 557-55

_

2X),

=

aK

_=

Xidy

_=

(6X Dy

(€6 G Xda

mo

ee

(0

~

K,G

¢€¢e™)

Suppose p

ts

@© a

its

elass

a

elass

on

on

Then

G.

H

CG

and

TSS

a

funetton

funetion

oea 5

(3.4)

= (ov,.)° eS

ey Proof:

Let

g €6G

_, then

Coy) (g) = 2

IH] x

= Tal

=

acxgx ty weg)

eee

@(xgx72)

= ial eee

= (8° We that

H

turn

is

normal

in

of

afforded

a character For

any

x

gxg

>

g €G is

posite

map

affording thus of

.

the an

the

We

map

analysis @G

=

denote

by

the

of

the

H

to

e@8(h) of

G

Ge

the

= on

case

Let

6

representation itself

of is

special

H26G.

by

¥(ghg +)

action

of

» written

character an

Oo

automorphism

y8(h)

defined H

H

an

(8,4) (xgx7*)

.

next

to

Wxgx7 2)

H a

.

given

Hence

be

Y

by

the

com-

representation

e(ghg~*) the

set

stabilizer

. of in

We

have

characters G

of

6

under

this

Since

6

follows

(3.5)

action, is

that

constant

that

Gy

Then

ts

an

H

trreductble

there

©

¢|o&=0} classes,

one

and

wW

Go

which

of

X ©

constituent

extsts

following

it

only

,

of

.

one

Xe

and

trreductble

satisfies

a)

(we,X)

#0

b)

(Wy)

HO.

tive

e , and

tnteger coset

Irr(G)

both

of

conditions:

Ww

of

{g

H? G

HOH

,

closed

under

taking

by

(3.13).

We

.

By

a ring

© Ch(G,#)

(3.14)

it

sub-

are

try-

suffices

3-13

to

show

there

that

for

exists

H

each €

Xx,

can

Po»

Xy

Let

P

let

H

write

x

has

be =

a

elements

In

particular,

g

G

and

such

that

»°¢x)

#0

XX

where

prime

Sylow

p-subgroup

=

P

of

H if

€ Ng ()

N

to

.

G

order

, and

of

9%]

N

H/

some

implies

|H|

power

X1%q

g

of

-

and

tIlie

in

©

,

G

then

that

(1,,)(gxg

g&G

,

a p-group,

p

.

«2

a =

is

for

H

p

= Ng ()

€

1

-1

~)

H

1

=

has

to

= >

H

.

prime

This

1.°¢(x)

p

prime

p)

order

gxg .

Xo

Since

of

each

(mod

order

Hy op

all

=

©

H

XP We

x

e

o_o

|u|

(1

gENn

H

) (x)

= (1)NGe) Let

WN=

N/

, then

cise

(3-3).

and

x

is

equal

to

the

on a set

of

cardinality

each Pp

is

However, a

H

is

p-element.

number

non-trivial

, the

(1p NG)

number

cycle of

fixed

By

of

a

ap"

Sylow

fixed

x

points

(3-4),

points

of

# 0 (mod has is

by Exer-

p-subgroup

Exercise

|N:H| of

=

length prime

of

WN

(1g) Nae

a p-element

p)

.

Since

divisible to

p.

by O

-

are

statements

following

the

Then

G@.

on

funetton

class

complex-valued

a

ts

®

Suppose

characters)

of

characterization

(Brauer's

(3.16)

equivalent:

a)

6

b)

0,

ts

a generalized

6

ts

of

the

subgroups

form

character

ters

of

G.

8

on

G

E

for

all

the

set

Let

such

of

§

that

all

class

where

as

elementary

argue

(3.177

€Z

and

subgroup

is

is

of

G

,

subgroup

all

class

generalized

E ®

ad

a

Linear

of

a

3

a

set

2a.-).

AG

ts

generalized

functions

=

characters

of

of

subgroups

6

G4,

ring

the

6p

X

elementary

the

be

elementary

where

an

be

R

Let

Proof:

aS

of

of

combination

B-linear

a

E

all

for

character

generalized

tsa

elementary e)

character,

charac-—

functions

character

G.

on

G&

G

Let of

of

Q

the

form

of

some

be

G

linear for

of

of

all

character i.

We

first

that

QCRCS

of §S

,

S

is

aring,

and

Q

is

an

ideal

The Q

is

an

YW

©€ Q

,

where

only

non-trivial

ideal and

as

S

.

To

see

is

a

.

Using

Z

and

As

subgroup

(3.18) °

this,

E,

linear

(3.17)

let

is

that

OES ,

character

(3.4)

we

of

some

get

ye = Za.(r.%e » a i

= = a, OA, 0, ) i i

By

assumption,

of

irreducible

is,

too.

ES

is

induced

of

E;

>

linear

By

So

6,

(3.11), from 95.

(3.3) By

1,

©

Q

is

what

of

virtue

of

want.

be

(3.17),

written

and

subgroups

hence

A385.

i of

character of

as

linear

an

some

integral

characters

of

subgroup

E,

are

of

elementary

ype © Q we

only

implies

Suppose

>

character

induced

yield

this

E.

combination

linear

irreducible

linear

All

(3.18)

, because

a

of

Es:

of

every

can

and

we

i

G

integral

is an

characters

combination

Subgroups so

in

write

€

elementary

of

assertion

that

that we

need

Q were

to

= R= able

show

that

S$

which

to

write

3-16

with

a.

€ Z

and

subgroup

H;

of

G@; G,

can

assume

that

QO.

can

written

be

induced But

linear

then

(3.3)

Hence, suffices

linear

to

By

(3.15),

of

the

we

form

subgroup

N,(P)

group

G.

EQ

le

complete

the

that

le

of

unless

G

may

assume

that

where

P

is

a

N

C

is

a;

©4

claim

that

that

that

is @G

is

as

a

Z-

from

proper

elementary -

normal

p-subgroup.

an

it

quasi-elementary,

eycelic

evidently

each

subgroups.

written

itself

we

of

(3.16),

characters

isa

Sylow

so

of

be

induced

»

elementary

can

proper

combination

proof

G,

p-

Let

elementary

sub-

Write

for

ax.

all

i

and

the

characters

of

6.

By

=

suppose

Z-linear

some

Inductively,

H;

Weed o+c + i>3 N

irreducible

We

a

in

yields

(3.19)

where

as

, then

of

holds

of

i.

from

PC

and

all

characters

show

of

for

(3.16)

combination

Subgroups

N=

to

a character

XL)

X,(1)

(76

=

>

1

= 1.

X.

are

non-principal

(3.1),

-

whenever

By

(3.1),

a,

we

#0.

get

Namely,

for

some

character

1

whence that

a, N

=1l1

normal

by

the

Frattini

P

and

=

p

for

, and

X, (1)

6

“aa

the

kernel

of

G

argument.

must

both

some

é(l1)

, and in

subgroup

then

=a, +

= 0

contained

proper

p8k

=

,

is

p®

6

be

Sylow

k €K.

=

, but

That

,

1,

-

K

of

this

is,

It

is

for

Then

gk

EN

X

>

a

impossible

any

p-subgroups

follows

g€&6G of

CK

K_

, so

,

so

geKk.

We for

all

(linear) (3.11),

conclude i>

that

1.

character, and

(3.19)

in

(3.19)

Therefore, A,

we have X,

G

= A;i

, of a proper

becomes

X,(1) for

> 1

some

subgroup

of

6G

by

with

as

€ Z

and

subgroup

H,;

of

8,

forall

(3.16)

holds

assume

that

05

can

written

induced But

linear

then

(3.3)

Hence, suffices linear

to

(3.15),

of

the

form

subgroup

and

= NQ(P)

eq

G.

the le

of

,

may

unless

G

assume

that

is

a

N

C

is

so

that

of

be

written

is

itself G

a

is

subgroups.

it

as

a

from

Zproper

quasi-elementary, normal

p-subgroup.

an

of

elementary.

cyclic

evidently

we

each

(3.16),

characters

is

Sylow

»

elementary

can

proper

combination

proof

induced

where

» then

of

le

that

P

p-

Let

elementary

sub-

Write

(3.19)

where

yields

some

Inductively, Hy

Z-linear

complete

PC

in

from

G

we

a

of

i.

characters

show

of

By

group

as

combination

subgroups

N

to

character

G,

can

be

a

1l,=a

a;

E€4Z

irreducible

forall

i

characters

of

and

G.

the

By

Xs

are

non-principal

(3.1),

a,0 #2 (9.35, = (este = 2 N’"G’°G N?"N’N We

claim

suppose

that that

X.(1) X,Q)

> 1 =

1.

whenever By

(3.1),

a; we

#0. get

Namely,

for

some

character

6

,

and

then

+

whence that

a. 1 N

=

is

1,

contained

proper

normal

by

the

Frattini

P

and

p8X

-

P®

p

for

620

,and in

subgroup

the

of

must

both

some

X NY

, but

That

be

Sylow

k €K.

,

=

kernel

G

argument.

6(1)

1, N

.

K

of

this

is,

It

is

for

Then

gk

x

>

a

impossible

any

p-subgroups

follows

g€&6G, of

© NCK

K_

so

:

so

gexk. We

for

all

(linear) (3.11),

conclude

i>

that

1.

(3.19)

Therefore,

character, and

in

(3.19)

rs

,» Of

we

have

X,; = rS a proper

X.C1)

for

>

1

some

subgroup

of

G

becomes

1 G = (1,08 -

=

a, #0

are, 1il

Oo

by

Exercises

(3-1)

Let H

84> of

8

be

Z

G.

class

(3-2)

general,

subgroup

H

the

function

class

{x1 5X5»

9®

sees

a

N46,

function

on

Then

on

H™

be

6

H/N

H

for

a

on

Suppose

G

F-vector

space

acts

obvious

way.

by

module

and

let

(H,K)-double

€G,

of

and

constant

to

obvious

a

is

on

cosets

class

way, this

®

a

class of

function

and

every

form.

class

Moreover,

G/N on FR

If

be

kK

is

is

ef = &

o~

yoink

which

H/N

a

Then

CH

an

on

© H*

G,

of

ae

te)

6,8.-

by

y

of

set

GAG

#

let

given

corresponds

in

€G,

t /,*i 8

N

—

this

subgroup

function

x

subgroup

x,}

Suppose

function

(3-4)

For

on

G

(8,85)

1 O(xyx")

6,Ye _F

(6

€

a

:

class

representatives.

coset

N.

a

G.

= is

G 85

1 +

be

of

K

G

however,

Let

e*(y)

(3-3)

6>

(Mackey)

Suppose

on

Then

(8 1 + 8,) GL = In

functions

a

finite

becomes @

then

is

@(g)

the

set an

9.

FG@-module

character

is

Then

the

the in

an

afforded

number

of

3-19

fixed

in

points

2

and

then

(3-5)

6

Suppose

Viewing me M eo,

=

of

g

H (1)

is G

as

= M Onn

FG

affords

the

2.

If

G

stabilizer

is

of

a

transitive

point,

.

HCG

FG

on

and

an

M

a

right

FH~bimodule,

becomes the

is

the

a right

character

6,

FH-module

abelian

FG then

module. u&

group If affords

§4

Splitting Given

let

RL

Every

Fields

a

ring

denote

R

o.:

RL

of

natural

isomorphism

another

way

In

section,

this

happens

we

of

F

following

Suppose

JE:K] regular By

the

on?

5,

©.

that

are to

The

$6:

matrices R~+

S

=R

nm to

restrict

above

,

over

has

a

there

R.

natural is

a

(this

is

just

can

be

partitioned).

matrices

going

n

investigate attention

remarks

are

what

to

proper

relevant

in

way:

K CE

=n


n

the

anda

the

homomorphism

extension

and

are

Then

E

representation above,

there

BE, > K+

For

subfields

is

a

any

F

, with

a K-algebra

yields

is

of

and

an

embedding

natural

extension

a © E

, we

the $:

define

the

E

>

KL

trace

map

trp, 6)

ff

ME

ED

we

leave

it

(4.1)

tro, Qt) )

Of

interest

particular

=

to

=

is

tr(o(a))

the

.

reader

to

trp, (tr(™))

the

case

that

verify

that

:

E

is

Galois

‘

over

K

mial

f(x)

.

Since

Here,

choose

over

f(x)

satisfied

K

is by

, of

the

acteristic

polynomial

s

=

(a)

is

transitive

are

is

in

E

f(x)S

On on

,

it

the

the

is

,

easy

K|

K

and

is

certainly

f(x)

is

the

minimum

of

of

$(a)

other

hand,

to

root the

char-

minimum

poly-

that

all

that

the

Cx-07)

=

the

£(x)°

because ,

.

the

is

f(x)

see

of

implies

of

polyno-

|K(a):

f(x)

roots

minimum

=

every

root

of

,

where

Gal(E/K) of

which

polynomials

and

o€Gal

have

a

t

over

Since

polynomial

.

with

9$(a)

.

irreducibility

|E:K(a)|

E

degree

matrix

of

characteristic

©

irreducible

polynomial

nomial,

a

the

same

(422)

roots.

(E/K)

In

particular,

PEK The

foregoing

Suppose

that

6

Since

{¥(g)|g

there

are

©

G}

extension

E

a

character

of

Y

,

and

K

, all of

applies

is

only

a

their K.

o

coat (EK)

discussion

F-representation

Soe

that

finite

G is

number

entries Let

to

96:

lie E+

K,

characters.

afforded a

subfield

of

matrices

in

some be

the

by

the

of

finite regular

F.

representation,

and

Then

Kam

bo %t

G+

By

(4.1)

the

is

given

by

(4.3)

we

of

K

want,

,

in

is

we

=

can

which

tr

in

K

may ,

in

happen

Y

»

of

this

(@(g))

E/K

, g

E

(4.3)

may

the

we

m

of

G

representation

Galois

be

z

case

degree

EG

toa

extension

rewritten

as

ef

o€Gal(E/K)

that

has

K-representation 8

case

which

that

enlarge

0, = It

a

character,

8, ¢g)

If

suppose

values

write

69

of

6

already

> and

EK

lie

(4.3)

becomes

8, Let's

summarize

(4.4)

Suppose

6

subfield

of

the

extenston by

an

such

=

|E:K|@

discussion

is

.

of

Then

K

character

E

of

,

a

the

of

there

sueh

E-representation. field

this

a character

F

E

to

point.

G@

and

exists

that

6

a

tis

Furthermore,

funetion

trp), (0)

K-representatton.

K

isa

finite

afforded for ts

any the

By ter

a

of

such

K-trreductble

an

a

irreducible

character,

viewed

as

an

irreducible (8,x) o €

=

=

8)

is

of

yy.

tuent a

+ 8)

a

of

other

no

Furthermore, 6,

eb)

of

(and

hence

»

)

a

m=

(6,X) =

such

sum

0 that

@

m8 4

=

6

character

constituent.

can We

have

proved:

(4.5)

subfield

bea

Let

K

Then

there

such

that

exists the

a

of

F

positive

character

and

X

integer

,

consti-

particular,

In

.

e@

copies

e

of

K-irreducible

a

,

of

constituent

therefore,

as

isa

K-representation

K-irreducible X

85

e

K #4.

to

representation

K

Y

and

X

(8, 89)

direct

to

similar

is

VY)

the

of

integer

every

an

get

K(X)

Let

where

the

when

is

, we

by

, then

irreducible

of

course,

X

K

x°

is

charac-

automorphism

€kK.

285

Of

that

values

some

of

that

follows

It

the

reducible

the

positive

a constituent

constituent

have

6= m6

Since

®.

affording

Yo

8)

an

be

Y

Let

.

is

be

Denote

and

character

the

is

28 4 Yo

there

mean

9 ©

any

z

write

(4.4)

for

o€Gal(K(X)/K)

then

can

(8,x°)

8

may

Since

adjoining

=

,

Suppose

by

of

By

6

(2.2)).

constituent we

it

(see

obtained

Let

call

constituent.

(0,x)°

we

K-representation.

F-character.

Gal(F/K)

field

character

€

Irr(G)

my 6X)

is

my (X)

The X

of

character

of

index

Sehur

a

K-representation.

@

=

the

ts

68

and

,

KCF,x€Irr(G)

Suppose

(4.6)

).

observe:

next

We

K.

over

the

called

is

my, (X)

integer

X

of

choice

suttable

(for

form

this

of

ts

character

K-trreductble

every

Furthermore,

representation.

a K-irreducible

by

afforded

is

)

o€Gal (K(x) /K)

Then

my (X) | (X58)

Write

Proof:

be

chosen

so

notation

can

Q0

for

(8, »X)

and

= my, (X)

(6, »X)

that

(4.5),

By

characters.

K-irreducible

tinct

dis-

are

8;

the

and

integers

positive

are

a;

the

where

aon

+...t

a8.

+

a,o,

=

O

im>il.

(4.7)

KCF

positive

tnteger

afforded

by

In

Let

Proof: —_—_——

By

(4.5),

8)

isa

,

the

my (X)

K.,

1

=

K(x)

nX

character

ts

m,(X) [n

iff

a K(X)-representation

words,

other

n

any

For

.

X © Irr(G)

and

Suppose

= Me (x 6%)

and

K-irreducible

let

6,

1

=

my,

character,

Ky

(X)X

and

@ = m,(x) is

a K-irreducible

}

character.

By

my X18) On

the

other

hand,

6a == try is

We

afforded

(4.8)

then (see

My

my X )

(4.6).

HCG

and

Then

e&

ts

a

Proof:

If

6

is

We

is afforded

now

the

L

x Ke}

g@Gal(K,/K) whence

X)

by (4.6).

(X)

and

the

result

O

8

ts

K-character

afforded

by the

character

Y

K-representation,

that

Exercise

that

Ll

my

Suppose

e°

shows

m(X) | (8 5X)

from

follows

a

(4.6),

= my (X)

/4(0,) =_ me )

Ky /K

by

conclude

(4.4)

x?

oeGal(K, /K)

a of

by

KG

K-character

of

H.

G.

the

KH-module

module

M

M @q, KG

(3-5)).

turn

to

the

main

result

of

this

section.

,

(4.9)

Let

n

be

suppose Sylow

p=

that

2,

We

faithful.

Since

tation

We

can

show

that

for

some

integer

of

of

G

6.

mX

a

, and

6 = 8,

of

each

some

+ 6

w

on

|G|

.

that

an

xX

(4.7)

by

a

Fix is

we

that

X

may

€K

,

K-represen-

p)

character be

K Irr(G)

by

(mod

d

o€Gal(K(w)/K)

.

85

is

irreducible

(l0O. integers i “xy = t y x for some integers s

isa

group

root

, the

where

for

G

the

So

prime

pth

of

and

q=p.

hypothesis

= A

and

Hence

that

that

=A,we=,

= Fy

some

r|n

image

| NA| = p* ale s y =x .

of

Since

use

Sy

G/A

a primitive

cyclic.

Let

that

may

.

implies

Thus,

a power

homomorphic

therefore

t

= A.

is is

last

faithful,

follows

= K(P¥1)

can

is

& CglA)

xX(1)

K(A)

}

= ACexe ty),

of .

We

of

order

order at

conelude

most that

@& construct

a

K-representation

affording norm

The

X.

map

key

tion

E>

Let

that

affording

The

we

image

will f:

be G>

a the

F

Ney, Cw)

xS pth gate

is

= yP

root if

this

{.

K-vector

to

let

be

Ne/x

construct

norm

map

N.

root

of

unity.

for

the

a

of

is

-1

Our

the

an

order

unity.

necessary,

K-representa-

we

.

>

w

in

root

ACx®)

s

AGE).

space

map

that

E

such

that

is

also

by

a

that

=

to

such

unity.

may

-

the

of

w

assume

it

a

= ACxt)

Replacing

Npjyfw)

use

o € F,

element

p*

and

Because

exists

p*th

contains

representation

= ACy txy)

Cx)

primitive

has

a

E/K

moment

E

space

there

= AY

there

b-a

the

p°th

is onto,

(4.12),

and

K-representation.

~[Cx) By

need

of

assume

construct

K(A)

is

X

primitive

Let's

=

K

fact

(4.12)

E

Since

a primitive

suitable

conju-

4-13

transformations

aACx) aw

mS "

R

K-linear

=] KK N

Define

Then

X™

= 1

aY-

» and

.

i

=a

w

we gil

for

see

u)

all

by

i-2

X,Y

on

aeéEE

induction

---W

E

via

.,

on

forall

i

that

i

whence

for

all

a

© E

aY

D-a

, 1

=

so

ONG 746)

ar(x®)

that

yP

“XY

La?

ow

x® -1

NN

ayP

Cw ty?

Dea

-1

=

-

axS

Moreover,

ACx) 1% w

2x Cx) Fy

"

aacx®)

for

all

Satisfies

a €E the

aX

, so that relations

Ylxy (4.11)

= xt there

Since is

a

homomorphism

4-14

¥:

G >

X¥(y)

=

C GLCE)

that

is

just

of

that

the

character

X(x)

element

afforded

A(x)

by

same

analysis

we

can

Since

G,

SO

Hence is

it

obvious

order

it

follows

we

are

K¢

YT)

,

p°

This

extension, fields

lying

= Gal(K(“/1)/K)

implies

that

a power

of

the p

the

fundamental

the

set

of

set

is

of

theorem

subgroups

intermediate

of

ordered Galois

fields

field

with K

between

of by

some E

and

,

are KCYT)

our hypothesis Of

whose

inclusion.

theory

between

Because

.

for

K

is Abelian,

linearly

irreducible,

a €K

together

and

,

G

primitive

some

over

< p>

this

The

in

extension

, the

0

,

.

Xp

Since

and

therefore

, and

=A

oO.

central

for

GE

2x,

a>

x(1)®

the

of

(4.12).

assume

is

that

xt

x8

faithful

a>

that

x=

is

X =

shows

prove

may

X

a constituent

that to

represent-

if

element

evidently

a

degree

e.

any

we

is

unity

of

has

of

,

x(x*)

intermediate

Since

0

Since

.

to

is

that

(4.2)

APCx)

shows

yp?

=

assuming

integer both

= X

regular

so

EZ

remains

a=

that

root

pth

only

p*

has

(3.5)

for

x°

element

,

,

then

A

that

the

E

oO,

applies

conclude = A

©

%* ,

X(x)= The

X(x)

Y

Notice ation

such

K

index

is

Hence

implies

that

and

KCVT)

whose

degree

linearly

K

over

ordered.

contains

We

a

a

is

Ye

, then

our

bea

=

also

of

unity,

w .-

of

(4.9)

by

arguing

that

p°th

root

of

unity.

Let

for a-b

some

-1

i

,

are

root

= wS Pp

p

conclude

proof

Ney Cw)

of

we

pth

a primitive

w?

power

e < a-b

Since

conclude

is

primitive

Nps, 6)

=

K

w

1=0

integer

xt

that

k

, and

%

=

wt

where

2° So

it

suffices

dividing p

of

q

kP

pe

dividing

k-1l

on

-1

‘ is

We

are

p°th

root

have

a>

a= that

a?

show

is

induction

iL

to

+i pot

1

>

by

» we 2.

.

-1

the

Let the

exact

any

a

therefore Inductively,

be the power

integer K

to

i >

p=

have we

2.

k can

= 1

p

power

show,

p

by

dividing

0

a

primitive

a>oO.

if

of

exact

of

contains

where

hypothesis

power

suffices

it

that

exact

p°

, then

» for

unity

KET

that

, that

assuming of

= aX pe

i

garb

_k

In

fact,

we

Since

(mod

write

p*)

80

for

kP

some aL dL.

#0

>

2

hha divisible

dividing

kP

,

ie

(mod

p)

.

.

(pc**myP

po

The

m

‘

=

Since are

integer

+ p(pe*tm)P-t

all

by

but

p°

bit aie

is

following

the

.

»

porate

two

the

pra

powers:

‘

tee.

last so

Taking

p(pt*tm)

terms

exact

+ 1

of

this

power

of

sum p

.

corollary

Qo

of

(4.9)

will

be

used

later.

(4.13)

Suppose

that

where K

pny

be

a

Then

nt?

my (X)

=

We

have

Of = Gal(KCYD/K)

is

automorphism

is

group

p

over,

v-l © K

divisible

by

any

of

it

odd,

Since

by

exponent

, for

subfield

primttive

Proof:

the

some

odd

F

which

of root

1

for

K@VWL)

of

hypothesis.

primes.

its

prime

p.

and

X €

Irr(G)

to

By

both

a

VY-1.

a subgroup O

Let

so

group

that

Png

contains

= K(P*V7)

cyclic

follows

G

unity

all

isomorphic the

of

of

is

(4.9)

of

order

cyclic.

m(X)

the p®

More-

not

is Oo

Exercises

(4-1)

mx)

|x)

fields tation.

K

for of

all F.

x © Hint:

Irr(G) The

and regular

all

sub-

represen-

§5

Some

Commutative

In

this

section

commutative for

our

rings

call

subring

a

of

of

F

if

we

call

valuation

ring

if

of

R

is

Also, AP, R

the

P*%+F*/U

it

aU

check

that

v(atb)

then

R

a

v(atb)

is

of

facts

set

the

to

is

L

a

field

F

for

every

x © F-R

a

integral


030

na

also

so

if

we

that

immediate

set

P

=

{x

of non-units

of

R.

multiplicative

homomorphism

and

because

,

it

follows

the

a,b

set

min{v(a),v(b)}

let

linear

is

of

now

call

It

,

Suppose

We

then

a well-defined

, that

-

For

,

F

, then

PF,

in

of

fractions

F

of

additively.

if

is

of

of

of

subring

group

subring

field

a subfield

valuation

p=

has

pR,

can

, we

maximal Hence

Suppose we

RC

Uh

bound

>

p

R,

it

for

a maximal

(R.p) isa

order

and

ordered

is

easy

this

to

subset

element,

in

which

%

> bDER-p} .

§S ,

a linearly

is

an upper @

pairs

partially

if

Iyer UB

ordered

containing

pons

R=

(Rip)

unique

R,

isa

defined

exists

of

R= {ab™*|aeR , bER-p}

the

then

maximal

set

of


we

any

exterior

the

of of

Hence

the

of

a

product

all

all

Since

get

endomorphism > for

advantage

-

unimodular.

= d(at(B)) g.c.d.

I

g(x)

= g.c,d.(b; 4)

for

if

finitely

Namely,

o(x,)

is a

d(B)

be

that

must

View at

useful

elements

x;

units)

, let

a

principal.

units)

B-~A.

B

: operation

also

assume

the

define

language,

the

with

divide

d(B)|d(UB)

free

adding

such

that if

j>l.

Xyo+++s%,

» define

= d(A)

for

assertions

is

(b55)

d(B)

plain

R

subset

(modulo

and

we

(modulo

= d(UB)

0

B'~A

observe

must

then

d(B)

functor,

Now

first

x,

finite

j>l.

minimality

(and

, of

so

, but

for

matrix

the .

0

then

column

generators

degree

R-matrix it

we

Then

=

if

induction.

tool)

of

1j i,j

a

the

i,j

by

divisibility

Thus

then

all

ideal,

a

yields

uniqueness

computational

generated

some

contradict

immediate

To

choose

first

-

will

for

operation

i. ixi

fancy

In sub-

language

is

the

=

d.(E) 1

=

Since

5

detA

UAV

=

ery

matrices,

be

a

j=

I

is

a

and

-1,-1 Az=UV

so

U

and

is

a

up

to

all

may

take

V_

are

products

of

multiples.

then

we

product

that

particular,

unit

unimodular

itself

unit

In

point

and

i#fj

immediate

i.

determined is

A

for is

it

forall

e.. JJ

J

The

directly.

4.400

i.

all

for

,

B~A

then

i,

at)

that

follows

atcusy) = atcuyatcayatc)

deduced

with

Ces,

if

Finally,

4

course

uniquely

are

e;;

whenever

for all

esi leisi1 iti d.(A) 1

a, (A)

=

A-E

if

that

of

may

fact

This

=

d; (B)

that

it

functor

a

is, and that

U

is unimodular if so

is

ai

because

that

is

e;yet of

elementary

.

elementary

matrices.

O

§6

Decomposition

Armed and

the

the

main

with

the

results

prime

prime

a

time,

ideal

R

#

our

so

afforded

an

reduce

by

mod

P

to

get

interesting

information

to

in

study

is

which

this

section,

their

basic

however,

that

they

no

it

the

a

depend

on

the

that

ring

the

all.

To

effect

for

of

fix a

a

X

with

is

The

we

It's

can

choice

R

to

of

the

remainder

the

get

study

that

develop

of

we

set

use

of

a

at

In

few

of

the

outset,

non-canonical

in

particular

ambiguities, ‘we

chosen

and

following

of

we

characters.

are

avoid

is

and

noting

characters

be

then

mysterious

and

worth

can

x,

tool

Brauer

these

every

G

careful

main

a rather

called

So

a

roughly

(5.4),

group

by

out

fixed

By

irreducible,

about

arithmetic

Then,

which

to

rational

R*-representation

summarize,

the

%

down

numbers

finite

§81-4

carried

pZ.

of

Brauer

emphasize

in

PZ=

xX

properties.

we

algebraic

follows:

introduce

R.

be

on,

in

get

is

we select

reduces.

subring

for

now

longer

valuation

and

analysis

as

functions

we

that

study

is

is

phenomenon

complex-valued

this

a

the

an

¥*

way

which

R-representation

Typically,

the

now

that

strategy

character

we

(5.1)

such

developed

section,

The

of

irreducible

theory

from

using

subring

speaking,

last

business,

Next,

valuation

the

characters.

at

p.

maximal

of

of

character

of

order

properties one

Numbers

these

fixed,

once

notation

notes:

will

F:

The

algebraic

in

the

complex

A

fixed

rational

R:

A

fixed

valuation

of

R

The

unique

so

v: :

are

that

The

residue

finite

group.

(6.1)

Let

S

be

prime

to

all

p.

map.

elosure

Proof:

and

are

homomorphism R™

-

Every es

coefficients

.

will

roots

Then

S

group

of

roots

of

S

element @Z@

R

is

chosen

.

For

F.

R>+R/p

G

field

obviously

in

map

the

from

Elements

R.

of =

R*

always

into

te

tsomorphie

R"

of so

R

R

F

F

F™

of to

order

the

residue

an algebraic

lie

R,so

in

*

multiplicative

satisfies

every

of

the

ts

a

elements.

in

in

in

under

p

unity

the

subgroup

unity

of

units

denote

of

In particular,

of

All

of

multiplicative

multiplicative

class

.

integers.

ideal

letter

of

F

atperR*.

the

consisting

numbers

pa.

class

the

rational

of

R-valuation

a=

before,

local

PO@=

aER, As

subring

maximal

canonical

the

prime.

called

The

of

numbers.

p:

p:

(5.5)

closure

element

R

by

induces group

a of

a polynomial with % + ge

of

R

satisfies

a polynomial pres

with

pz,

algebraic

ues

w

of

€S

every

unity

p)

roots

mod

§

algebraic

We

=

the part

is

R*-{0}

can wt .

set

of

then

We

for

conjugacy

S(t)

is

called

refer

to

the

these

classes,

p-part

the

set

s(t)= U(S(m)).

we

of

is

.

[In

an

nth

However,

for

some

has

is

if

n

Z

0

distinct

a bijection

particular,

further

am

R”

is

if we

1 any

an

elements use

is

g

the

a

£

of

and

op

if

its

p-

prime

T

, we

of

@

lower

if

let

S(t)

to

7 .-

want

the s(n)

its

p-

that

TF we case

Pp);

such

G-conjugate

@ ,

to

p-regular

constituting the

g

p-element

unique

p-element

is

is

form

p'-element,

of

order

call

x

a

Any

the

p-regular

its

is

in

P~part

classes of

p

&

t-seetion

will

notation.

written

the

call

and

Thus,

the

p)

p-element,

call

Finally,

R"

polynomial

uniquely

a

otherwise,

1.

elements.

of

ww”

an

0

some

be

g.

p

is

2 .

is We

since

R™

X"™-1

In

(equivalently,

is

element.

x € k

and

of

1

p'-part

the

that

pt

p-singular

be

follows

introduce

p'-part

of

.

» but

of

(mod

, it

where =

#0

such

g © G

tp

field

root

of

z"

element

n

a

in

therefore

every

next

179

and

is

closure

element

the

some

since

p

between

&§

w

and

of

and

non-zero

for

, then

(mod

Z/pZ

extension

particular,

root

coefficients

if

to

union ,

of

thus

Now 9

©

s(l)

for

some

roots

nth

suppose

jf

.

560)

n

zZ#

0

of

unity.

roots

of

distinct) ve

Then

(mod By

F

ve

R -representation

satisfies ,

so

(6.1),

function

of

f

and

xr} are

uniquely

(not

and

nth

determined

necessarily

eigenvalues

of

£(p)

are

is

k

2

w.

isl

.

+

called

the

Brauer

character

satisfies

b(p) = tr flo). The

function

6

classes,

so

by

usual

view

as

a

$

similar When

the

is

function

obviously abuse

from

R-representations

there

is

characters

of

danger

then

recall

of pairwise

ations

of Let

from

from

number

G

of

of

have

, so

£64 2b52-++ ob}

G-conjugacy

notation,

we

will

the

Brauer

F.

same we

as

Brauer

will

that character.

refer

ordinary

there

are

irreducible

{fy ofooee+sfgh be

Note

also

to

characters

characters.

) that

non-similar

suppose

on

to

confusion,

( 1.8

constant

S(1)

F-representations

distinguish Now

set.

=

s(1)*F

(6.2)

to

G

define

We

96:

are

Wy

the

o(o)

The

eigenvalues

there

that

of

a polynomial

its

Wy Wares

such

»

Wy W1 Wo rere

te

an

p)

unity

in

ve

is

the

only

a finite

R*-represent-

is

a complete

corresponding

set

of

Brauer

characters.

Brauer

characters

(6.3)

The

trreducible

are

linearly

to

F

are

linearly

.

Proof: G

with

of

G

, then

all

the

character

them

characters

functions

over

R

Since of

mw

are is

matrices

upper

(x-1)P"

1.

Now

and

triangulized

Py

tk

.

am

=

f (1)

S(1)

G1 Po r2-2

&

0p

R -representation

If

a p-element

Br(G) .

from kok

is

a polynomial

xP.

f(m)

by

4 bg 20+

funetions

».

satisfies

the

denote

trreductble

a p-element XPM

for

over if

and

g ۩G p

Fp)

because

R™

and

is

p-

can

be

they

>

commute,

therefore

(6.4)

te(flg)) = tr(f(mf(p)) for

Now

let

Brauer

ds

all

be

oe

Sf

Cte

Use

c

>

=

0

= tr(f(p))

gEeeG@.

an

character ®.,

a dependence

lu Mer

is

~

there

Brauer

the

be an irreducible

where

then

we

independent

n.

Simultaneously

with

and

called

independent

f (1)

pt

regular,

G

f

Brauer

eigenvalues =

are

Moreover,

integer

§ = 7p

and

of

Let

of

some

These

irreducible

(1 with

By

the

representaEB 2S22°°°

By

orthogonality

have

(g74)y(g.)

restrict

> v([CQ(e)[)

some

conjugacy

representing

relations

=O.

numbers

1
0

x, 61)

therefore, a¢B viags) > vO C2)? + v(p"P?) ~ vce] h

= y(p 2) It

follows

therefore

[G]

an

= P°gy

ai5 = 445/P

integer.

where

g,

(7.15)

—

can

a.

be

a local

# (0)

mod

for

some

if

p

we

integer

and

is

write

, then

integer

f,

# 0

(mod

p)

rewritten

=—a

(7.16)

a...’

soa

To

d),

multiply

obtain

* is

Moreover,

= p a-d(B)th; if,

x,(1) and

that

h.

5

5.

(mod

p)

(mod p)

(7.16)

by

for

all

for all

p

h.

+,

i,j,2

i,j,2

.

.

7-23

h.

i

= mG =P

4i5

From

(7.17)

Pp

the

following

rlP i “hy

ij

a)

C,

B

block.

has

exactly

ty

e)

If

Proof:

characters h,

to

1 a,, 2p

pats)

J.

+

1 =

d(B)

that

all

characters

«However,

, which

is

of

if

d(x,)

impossible.

defect

one

= 1,

It

have

height

Zero.

oO

The the

preceding

Cartan

observe

results

matrix

that

become

there

is

them.

Namely,

from

X

the

of

is

set

character

table,

@

Brauer

is

the

on the more

matrix

p-regular D

interesting

a different

the

is

elementary

way

equation

columns

the

character

t.t xtx = Btptps

of

of

we

= Ftco

when

we

computing X

=

the

decomposition table,

divisors

Dé

where

ordinary matrix,

obtain

and

of

7-26

t

=

XX

since

R

(5.8)),

it

and

(see

are

{p

must,

by

C

of

these

with

coincide

As

way.

interesting

number

is

equal

On

the

matrix

Cartan

the

Oo

of

to

other

Then

.

the

number

hand,

by

divisors

elementary of

(7.18)

(7.17)

with

let

usual,

|6|

dividing

the

combined

can be

result

p

proved:

we have

of

dividing

(pl®) 1x E€ s(1)} .

are

of

divisors

,

d(k)

p

of

power

exact

Then

.

x, © k

elementary

Phe

the

be

to

,

k

We

C.

integer

an

k

class

each

for

divisors, of

divisors

2 CR,

since

However,

R-elementary

of

divisors

R-elementary

.

Ni

peee9P

Z-elementary

the

where

[CAC x9 |

This

1

uniqueness

of

defect

(7.18)

d

the

that

follows

define

therefore

the

d

divisors

elementary

of

theory

a

has

in

units

ane

Cc,

the

Since

columns.

is

¢

bya

¢@

from

=

(6.7),

By

.

e)

obtained

is

it

of

permutation R

Cy oP “Co oeeoP

since

unimodular

qn

d

dy

diag(p

ordinary

have

we

(2.12)

relations

orthogonality

"Cs

the

by

Then

0(1

b&

w(K)

that

interpretation

an

.

exists

there

(8.5),

by

, then

for

class

such

H_

k

exists

there

Then

Gy,

defect Hence

OH).

a block

wtth

that

such

G

a

be

k

in

6(k)

containing

We

oo

wk

contained

and

x€k

(xk)

groups

G

of

of

pa

6(b)

0

(b°)®

defect

of

defined.

is

pb?

whitch

p-subgroup

a

= W

a subgroup

ts

H

for

b

n H? )

correspondence:

Suppose

(9.3)

ab

b?

behavior

nice

get

also

We

= ua(k b

CG.

of

k

classes

all

(Ko)

vb?

for , we

8 (b°).

Oo

6 B

Thus

6

role to

= ¥

6,

B

later

This

on.

We

H

,

and

an

(6;x)xX

notion

next

characteristic

H¢G

Z

X&B

zero

"lift" as

ordinary

will

play

an

the

Brauer

follows:

Fora

irreducible

important correspondence subgroup

character

¢

of

define G

wSgy 2 tec°(1)in? G A

for is H

all the

,

(9.4)

classes usual

let

xa

be

Suppose ©

of

G

induced

K&

ts

the

H an

a)

b)

[| o> Cx)

, where

character.

class

CG,

of

b

itrreductble

we ob)

=

of

G.

If

v°

%y

& k= 9 In

G

ts

©€

For

RH, a

and

class

containing

a

block

character

w 6K)

of in

for

H

all

,

of

and Then

elassesHY, ,

Coe)”

then

k

k

b.

particular,

tis defined,

a

=

we

(w®* =u, C

.

p®

,

and

Cz

G

)_(1)

—

o°C1)

In particular,

0 l

if if

B#b® B=

b®

(mod

P

v((¢%),(1)) > v(eS1))

p)

.

for alt

blocks

B e)

of If

G, with equality iff 2°

ts

defined

Proof: By

Let

definition

%

of

GC

G

and

be

a

“HY

of

L

z

Vx

Ek.

Tay

we

have

|

zcly)

cox, Ik] oly.)

r°c1)

[S|

|Co(x,)| [G28] 5(L) k

Ken, thy zw

kG=K, follows.

with

G

;=

a)

G

Therefore,

we)

and

Ieg6y

To

prove

§

oly)

(xk)

b),

5

we

b&

ts

.

City) |]

|H] yexnH

| C650)|

then

b(exye)

gee

—

where

b®

characters,

o>

=

€

aclass

induced

OXy)

trreductble,

B=b®.

write

g

CCX)

*

xyek.

1%!

Gra

w~ (%) b

x x

r°¢1)

wi

for

any

w~

G

block

x

B

of

1

Ce.)

XCxy)

= (8x) xX) wo

o7(1) Hence,

(c®,y)

6G,

2

Say G

xX

Xx

(tf

G

z

5x)

xl)

w.Ce,)

(c®,x) x2)

(2%) 31) o°¢1) If

p°

is

follows

defined,

The

= Wag

is

an

question

of

exactly

mysterious. which

P-subgroup

P

obvious

covers H © G of

a), and

G.

UG HYP

all

we

a = X&

(4)

of

is

b)

a).

CQ(P)

to

4 H

aclass

of

:

seems

a sufficient

interest C,(P)

C

defined

obtain

of

that &

b&

can

cases

such For

corollary

when

However,

suppose

by

°

c)

condition Namely,

cae)"

immediately.

Assertion

a bit

then

us.

for @

?

some

define

Because

C,(P)

conjugacy

classes

linearly

(9.5)

JH,

to

The

and

a map

map

AOAC (P)

is

therefore

Z(R

x

G)

UGoH,P?

+

Z(R

*

@(R *

G)

aunion

Ye

3

of

H

H.p

extends

te

. ts

3

H)

+

2(R

H)

an

algebra

homomorphtsm.

Proof:

This

consequence

of

result

(8.1).

w(x)

extend

ul acR*e)

x €

G

,

suppose

=

For

to

UG Hyp

then K

write

=

by

uh)

a

x € 6G,

a further

define

x

if

x ©

C(P)

O

if

x

CCP)

a map (x)

slight

abuse

+

(H)

for

RH

.

x-u(x)

of

Then for

notation,

all

we

can Now

ACG.

subset

classes

i init = UGE) + OGIO)

oH, 6H,)

+

=

any

conjugacy

wG ud)

€

RG

Put

are

K,

essentially

=

linearly

+

and

is

of

G.

Then

+ O04.)

+ uGired,) + ¢@pudt,

If

x€cCCP)

and

y €CC(P)

then

xy

€

C(P)

,

so

that

uuGf,60f,)) = 0 = uo udt,))

BN

3

Now

uh uh)

il

za

therefore

rc

and

for

x ©

C(P)

+ ul oH,))

, let

Sy = Uy sy.) ly, EX,

- C(P),

¥z¥,=x}

Is, |

(8.1).

However,

= 0

(mod

p)

by

u(oH6H,)) v2 SO it

follows The

map

that

sufficient

ui.)

u

homomorphiem.

=

is

= xEC(P)

.

(s_ | “x *

= uA, uH,) known

Using

it,

we

condition

for

the

as

can

Then

the

‘

Brauer

establish

existence

O

the

following

of

the

Brauer

CG

and

correspondence.

(9.6)

Suppose

P

ts

a p-subgroup

PCg(P) CH CN,(P) . atl

blocks

Moreover,

b tf

of B

Phen

H, ts

and

a block

of

b°

ts defined for

WG

= O°UG HOP”

of

G

then

B=

b°

tn

which

Proof: H.

kn

CQ(P)

the

former

case

Let

a

of

for some block

If

k

uw is

=@

or

case

Vey >

=

Uc

Hp

of

H

iff

z

=

and of

CQ(P)

have

b

let

H

,

=

then

0

P & 8&(B) G

.

bbe

because

w, Ck)

*%

e

b

gtis

aclass

kC

we

5

ds p‘@p?

a block

either

C,(P) by

JH

.

In

(8.6).

Therefore,

for

all

homomorphism For

an

H

by

any

idempotent

of

G.

(9.5),

block

of

B

Z(R

b

*

of

H)

Then

some B= *

W,° Hen)

zero

(possibly b

G

for

=

1

iff

images

independent iff

there

nH) by exists

of

=

.

a class

of M4

* ey

}

b,¢s

set

class

definition

te ule)

, so we may

block

b€ES

uw

is

an

algebra

is

evidently

is defined. 6G =,

empty) some

Since

GS

* ulep)

for

= wren (A)

= wp, CNC. (P))

= wp (ACH)

classes

A

a

_

om

“A

G

wg?

S b

write

1

of of

blocks H

of

iff

However,

the

sums

of

G

and

therefore

, of

G

such

H.

set

is

of

non-

linearly

that

S

#

@

a, 4)" By

#0

(8.5),

other of

# ui)

.

ap A)

#0

hand,

B

if

Pc

ts

P a

ap (k)

G,

then

If

P=

all

blocks

u

=

follows

In particular,

ap(k)

(9.8)

can

To

now

.

On

the

class

.

p-subgroup

for

O

of

:

then

% ep

-

Since

of

from

= 0

G,

use

elass

=

we

the

(9.6)

unless

b),

some

CQ(P)

definition

obtain

first

@ p-subgroup correspondence

of blocks set

#4 uA)

& (%).

C

and

6(k)=P

have

Be

=

B

=

ep -

Brauer se

that

and

of

Ae ap lk) k

zy

uCe,)

k €C,(P) a)

k

%

, and

(8.5)(b).

prove

(Brauer's

the

n

#0

#0

UG .G,P

it

We

a defect

§(B)

correspondence,

proved.

is

&

kK C

by

is

, there

P

G.

B

a)

S 6(B)

a. %

Put

iff

6(%)

a normal

If

#0

if

ap A)

of

b)

Proof:

ts

uA)

aA

block

a)

for

6(B)

satisfies

Suppose B

only

G

which

(9.7)

Clearly,

of of

main

of

G.

Then

a

bijection

ts

NQ(P)

blocks

theorem).

with of

G

Suppose

the

its

Brauer between

defeet wtth

P

the

group

defect

P

group

set

and P.

.

9-11

following

the

establish

first

We

Proof: assertion:

k

Namely,

if

and

X,5y

P,Pp&

€

p-subgroup

P,

P,

so

1

Cc NQ(P)

(9.9)

that be

is

is

put

b

is

and

H

defined G

#

6(k)

.

#P,

P

Therefore,

there

exists

, but

then

5 (k®)

a

a

P

and

=

for

and

and

P

As

w gtk” b

by (8.5)(a). G

4(b-)

H

of

=P.

b

6(b)

with

an

, then uk’)

=k

a

= wi (u(k™))

By (9.3),

suppose

> and

= UG Hyp

> HW

= Ng CP)

class

6(k°)

thus

P

a block

a defect

8(b°) CP

with

Cc CQ (x)

let

proved.

Now

yields

then

Syl ,(CgCx)) >

PE

If

xEk.

Suppose

. and

= P

6(k)

with

NQ(P)

of

a class

is

k

.

x8osy

and

Ng(P)

of

is a class

4C,(P)

= p

p&°

theorem,

ge © Ng(P)

whence

c © C,(y)

, so

g€&G

some

for

= y

Sylow's

By

.

Syl (CQly))

Therefore

x®

P

6(A#)

with

G

of

aclass

, then

€ ANC, CP)

some

for

is

M

6 (k°)

é(k) =P,

with

Ng(P)

of

each

For

.

NQ(P)

of

class

i

class

a

is

CQ CP)

AN

then

z P

6M)

with

G

of

aclass

is

K

rf

(9.9)

=P.

Let of

application From

A

= uw (k)

(9.6),

#0

k (9.9) b°

, whence

P= 6(b) € 6(b*) G

9-12

Next, then

u Cen)

blocks then

that H

G

with

B

Pm

ey,

é&(b)

by

so

that

6(b)

Brauer

with

Suppose

= be

e,

the

get

sum

1

ranging

=

6(b)

b

= far

the

group

P

onto

the

6(B)=P

,

exist

is any such,

5 6(B)

maps

set

set

P

by shown

of

of

blocks

blocks

of

P.

show

By

(9.9),

.

If

with there

correspondence

that

blocks (9.7)

bg

using

and

so

thus

en

But

(9.6)

b®’=B. P

G

have

are -

hy

of

We

to

bj,b,

bs

#0

=

group

remains

a block

(8.6)

defect

defect

It

is

te

b°=B H with

of

PC

the

of

=

b

(9.3),

and

suppose #

1

this

=

the

of

we

map

H

is

with

1-1

6(b,)

=

6 (bo)

have

z 6(k)=P

ay,

equation

A

1

(k)

can

A

k

be

2 a, lnc,(P))* 6(K)=P Py —

Ea, (uD) Py S(K)=P

WOH)

over

G.

classes

of

written

re ,(B))

However,

we

now

=

P

us

le.

Do

by”

)

=

=

In

order

to

next

relative

to

normal

correspondence group

P

and

Suppose, representation

X Fh) H the

over

=

ordinary

ued)

(e.

Do

study

)

then,

that

wo

1

dh)

oO

mileage

We

of

PC,(P)

any

It

Brauer

the

then

get

N,(P)

main

a

further

with

defect

. If

X

field,

isa

and

g©G

a representation follows

that

characters

of

formula

v8(h)

first

correspondence

H@G.

over

field.

from

Brauer

defines

the

1

=1

blocks

of

and

by

subgroups.

H

2

Pi

the

blocks

same

Py

uh))

Do

.

between

of

w

a GH)

more

(ghg 1) the

wh)

EF

get

we

argh)

by

(ui)

Py

&(A)=P

=

theorem,

a K))

Ea

=w,

by

EF

Pa

g(K)=P

>

therefore

a

s(n)=P

-

and

az

gtKy=P

= w(ghg7t)

.

G H

,

then

%©

of

acts

on

via

the

9-14

Moreover,

xX

this

€ Irr(—H)

action

,

g&G

preserves

blocks,

because

if

and

X

=

Isc1)

x

d..

geBr(H)

*¢

then

g x" lsc) so

that

d xe

numbers. also

any X

a

48

=d

So

if

b

block

of

H.

element

go € FH

€ Irr(H)

= 3 dys ¢ by

is

a We

block let

decomposition

H

,

then

{g€G:

b®

b&=pb}

is

.

, then

calculation

For for

yields

-

therefore

all

blocks

that

of

o° g = g 1 ag

, a straightforward

b

are

of G,

, define

er S

for

uniqueness

Xoo

g

and

g

geéG. of

H

uniquely

If , then

.=e

p&

{bj »--+- sby} clearly

determined

blocks

is

Ss 2 e,.

i=l

“i

a

G-orbit

©

ZCRG)_

B,,...,B, t

of so

of

there

G

such

9-15

In

this

situation,

we

say

that

Bs

covers

b;

for

all

i,j. (9.10)

Suppose ts

a

HAG,

block

statements a)

B

b)

If

b

tea

block

Then

the

of

G.

are

equivalent:

eovers

then

constituent

e)

b.

If

X€&B_

of

There has

Proof: characters

To of

show G

as

B

X,

a)

ts

Xa

in

a G-conjugate

an

constituent

in

X©B

trreduetble

that

irreducible

has

exists an

and

following

every

then

trreductble

d)

H

b.

X © B_

of

of

implies

characters

of

such

b.

that

Xy

eonstituent

in

b), FG

,

we

view

all

so

that

in

particular,

XCeye)

Suppose

HH,

B

that zoe

{bj 5+-- bg} By

are blocks

=

Sx x!

X(1)

is

a G-orbit

of

G , and

of blocks

of

b.

9-16

s

ze

tel with xy

b = =

b,

> B=

z r€Irr(H)

x(e, By )

=

Pi

the

that

.

i

remains the ze

x €

B

.

: Since

xe

°p.?

t

=

x(1)

, we

x(1)

= 2m,

The to

=Ze

i "i x ©

4

B

then which

(9.11)

and

write

=

x(z

and

show

that

of

blocks

B.

5

for

Xy

has

=

XS

cEUb, i

2

tle,

G

m

)

bs

o(1l).

Xo

x(z e,.)

#0.

i

implies

Suppose ta

B

=

b)*c)?d)

d)@a).

=

containing

b

.

Hence

of

B.

irreducible

xCe,

follows

are

in

so

By sbos+++

ob,

be

Then of

J

G.

If

constituent

j

it

) #0

for

in

some

b

j

,

;

.

0

HAG,

a block

blocks

it

trivial,

b

certain

Bs

are

Let

an

J

obtain

2

implications

=

en)

J

are non-negative integers, XE irreducible constituents of Xu

G-orbit b.

Be

m

all

Vb,

jer

Let

My,

xCe.)x

t

Ze

By

=

Since

=

b G.

te Then

a block B

of

covers

H

and b

“tff

B

the

funettone

restriction

ZCR

*

for

Proof: G)

#

NM Z(R

uniquely

the

e.

and

the wple)

algebra

follows.

a

is of

a

Bs

are =

1

must

can

be

B=

b;

H

idempotents

of

, it

g€G

same

the

algebra

, because Z(R'H)

Now

let

Thus

b

by that

follows e

isa

iff

n Z(R'H)

Z(R'G)

= By seee gb,

be

, with

of

G.

Bs

for

which equal

the

write

Goorbits.

H

blocks

of

of

of

of

$e

NZCRH).

all

for

of

aiff

G)

have

idempotent

G-orbit.

blocks

Oy,

blocks

union

homomorphisms

idempotent

an we

e

=

idempotent

{bj s+++ sb}

where

be

primitive

e& is

and

$e

Z(R

, then

the

{bj 5-++ »bo}

G-orbit

e

H)

Since

primitive

to

determined

are

(7.8).

a

Let

Wp

agree (see

Then some

on

wp le") j.

zl,

Since

two

a primitive

(1.8)(c))

the

result Oj

9-18

(9.12)

Suppose

Co(P)

P

ts

© HIG.

covered

Proof:

by

B

b

of

H

, we

have

we Ck)

k

is

aclass

of

G

and

thus

k NCCP)

Hence,

Then

= @

w,

(9.13)

for

every

block

and

any

p-subgroup

class

= up, 0 not

of

B,

by

wy (Kk)

=

now

to

the

of

G

the

block

However,

in

H

= w,

Brauer

te

contained

(9.11).

0

and

H

cover

of

contained

G

G.

both k

of

block

in

if

, then

Ck)

1

=

Returning easily

Then

a unique

Suppose

H.

a normal

by

(8.6).

Oo

correspondence,

we

obtain

Suppose

pb?

HIG,

ig

b

defined.

a)

b°

b)

vb’?

ce)

ts

a

block

of

H

,

and

Then:

covers

b.

te

defined

to

b

pi?

= v®

iff

for

b'

b'

te

G-eonjugate

G-eonjugate

to

b.

Proof:

character there

If

¢

is

in

b

, then

exists

X

€

b

G

an

ordinary

(c®) chew with

(X55),

irreducible

#0 #0.

(Wye) 49

by

(9.4) Hence

so b®

that

9-19

covers then

we

b

by

put

¢

= wee

t

(9.10)(d). =

and

cB

If

Ee b.

Since

therefore

p& .

Finally, if

then

b®

b'

by

= b® G

b'°

is

G

for 'G

=6&

some we

defined

and

equals

for any block

a)

so

b'

and

gEG

get

.

b'® = b°

covers

!

b

t

of

are

G-

b

b

H, O

conjugate.

can

We

(9.14)

now

Suppose

P

a block

of

there b

Proof:

ts

a

G

wtth

exists

yevee sd,

and

(9.8).

extend

Of

é(b,) By

a

exists

defect

assume

there

so

by

(9.13)

there

is

a unique

By

(8.6)

and

(9.3)

we

get

a block

PC, CP)

this

is

theorem,

analyze groups.

of

essentially we

the

defer

may

the

structure

b

of

wonder

fhen

of blocks

subject

blocks

PJIG.

PC(P)

with of

By p°

such

=

oO

what

of to

B

blocks.

=P. happens

defect

question

of

is

be = B,

that

G-orbit

6S(b)

with

the

B

1» we

we have

X61)

~

1)

k

of

the

p-regular

all in

the

x»

x

block

same is

actually of

elements

p-regular

the

)

Hence

G.

map

the

p

G

get

~

1

zr

=

You

~

TQ) xes(1) P|

—i

g(x)

PCD

z

g(x)

pCx)

© B-

so

Pl Tey xes (1) for

(mod

are

all

9¢,y

characters

Brauer

-=

[P

Plo

that

rg = (PIT, Cp

Finally,

as

C,

=

|P|Cg

=

[P| cE

, (7.13)

‘

[|

=

|kj|

(7.8):

for

holds

since

between

Xo

X12Xq

so

G

by

IK] x5 O%)

classes

Moreover,

G.

a bijection G

B _

of

k

in

congruence

corresponding classes

and

Ik] x, Cx)

p-regular

all

G

that

» and

G

of

between

k *k

a bijection

induces

> Xe

X%

map

.

implies

that

and

B

po CB)

=

then

|p] pt

there

that

a

w

#0

X

(k)

-

is

an

#

Let

k

0

Gnodp

)

.

Since

|6(B)|

Suppose te

Then

P

ts

=

|k|

such , we

get

G

= pa'B)

B

Xol scr)

= 4

all

elements

tf

{1

=

c-(7)

*

characters

y,(p) 0

°

p-elements

p €C(n)

a

.

tf

EP

tf

wEP

of

Then

G

irreducible and in

character 9S.)

are

of

and

P.

group

kernel,

its

Cg 2bz2%%+

trreductble

6 = PC, (P)

ordinary

Brauer

unique

the

get

Oo

of

tn

P

we

.

a untque

with

(8.5),

defect

wtth

G

of

Xo

Moreover,

ordinary

= S(B)/P

econtatne

ta

by

a p-subgroup

a block

character

for

|k]|

B,

= &(k)/P = 6(B)/P

§(B)

X;(mp)

yx © B_

for

so

(8.5).

B

character Since

G

(9.16)

class

irreducible

$(B) CCc 6(k)

by

bea defect

B.

the

p

,

define

(0 yCjPy:P]) Let

@€

Irr(E)

, then

and

that

we

x XEE

xX, Ge)

6 Cx)

= pEQ

x; (1p)

OCip)

Zz peQ

t.¢m)

X¥g(p)

6

a

TEP

Crp)

.-

1weEP

Since

E

is

irreducible

Hence

we

x

x(x)

xGE

The

m

m

is

(x,;:6), character

=

xeG

6G)

=

by

|P|

by

also

is of

Xi

— 2

such

Q

ain its

X Se)

pEQ

factor

.

of

The

|Q|

product

that

6

Py

of

two

(resp.

Exercise

(See

kernel.

an

integer G.

Bote IIL

the

above

in by

and

factor

[P,:P|

therefore we

¢3¢m)

¢,(m)

product

addition

Moreover,

—

~

TEP

left-hand

and

divisible

x-(x) x. CR) 7

)

6

Py

is

(2-3)).

have

divisible

sO

65

Py

(resp.

right-hand

divisible

product,

characters

Q

J has

8g

a direct

is

is

v(m) -

We

xy,

8,

1

an

an

(1d.

integer

integer

> v(|P,:P]) conclude

is

that

a generalized

have

63 Cn)

X9 fp)

X fo?

Tv

0€S(1)

[2

¢.(n)

| P

G|T,

nep

t-Gn) 10

i

-

f=

pes(1) *9

(op)

¥,

0

pI

(9.15)(d)

we

(X45 2X5

=

conclude

that

get

1.

|P|T,

Since x,

€

= TR=(1),

X31)

=

Irr(G)

6301)

and

so

that

Xf)

that

>

dy

we

oO

By

~

6301)

.

Thus

But

since

Cz

=1,

{Xp Xz o%%2 Xe} characters

In

character

of

This

the

|P|

set

of

by

(9.15)

ordinary

and therefore irreducible

B.

O

of

course,

customarily

we

have

referred

to

Xo

block is

this

=

ad. the

B.

X9

-

canonical

the

which

naturally

section

by

most

important

contains called

the the

giving

block

is

principal

prinetpal

character. bloek.

several

of

its

Suppose

HOG,

CQCéCb))

CH.

block of

of

H.

G

b Then

tff

b

ts

a block

of

H

b” ts

G

,

te the

the

and

prineipal

prinetpal

We

important

properties.

(9.17)

This

\

applications

block

conclude

the

particular, is

usually

is

in

character

In

Cy, =

block

Proof: and

the

Tf

k

result is

without By

Let

P=

is

of

G

points

so

(resp.

by

)

),

then

Wp

(k)

weby =

“By °

H

be

the

=

|k|

of

H.

and be

by =

p'S

=

that by

with By

b'

by

while

if

inductive is

.

By

the

P

acts

=

for

b'

principal character.

then

on

|k MH]

all

H=G

assume

block

P#1.

k-H

(mod

p).

of

classes

G xk

that

b®

we

may

1

with

H

is

the

=

Bo

for

assume

that

d(b,)

principal

=P

and

first

be

, then

6(b)

Let

(resp.

, therefore

= By b

is

L

and

the

6 (b, )

> P

, the

assumption.

character,

and the

first same

But to the an and

now

block

b'

Hy

-

of

By

PC,,(P)

>P

paragraph

to

.

principal

of

by

this holds

implies block of the

show

Put

block

conclusion

N(P)-conjugate

CG

and

of

PC,(P)

(9.13)

Hy

suffices

6(b,)

contains

if

block

By (9.1),

it

principal

thus

some

dCb)

=P.

paragraph,

(9.8)

contains

>

H , there is a block

then

N,(P)-conjugate

Hence

may

is the principal block of

6(b,)

NQCP)

we

principal

of

b'?=b

= price?)

If

by

applied to

PC.(P)

|k|

suppose

a block

» then

(9.14)

then

Inductively, is

By

So

P=1

°

Conversely, b

If

trivial.

aclass

fixed

&(b).

(9.3). of

proof, by

our

that

b’

of

PCCP) .

the principal O

=

(9.18)

Suppose

for

every

the

prinetpal

Proof:

by

by

the

=

by

by

By

(9.10)(b).

a

&

constituent

be

block

and

of

Bo of

we

Since

By

N C ker because, some

principal

N

x.

when

N C ker

character

wW

y

in

G.

N.

get

IN|.

Brauer

of

the

(9.10)(d), Then

@¢

p/

block

By

G-invariant,

characters is

Let

-

and

ordinary

principal

{1,3 is

NAG

covers Gker

x

6

for

all

by

(6.7),

block

of

d(by)

=

by for

+

6G ,

0

Since

all

x

© B

0

Brauer every

such O

10-1

§10.

Generalized In

this

Decomposition

section

we

obtain

Numbers, what

is

single

most

important

result

in

second

main

theorem.

Recall

that

blocks

as

decomposition

of

the

space

Our

is

now

to

a

functions. tion

in We

objective

a natural first

way

introduce

of

representatives

w

of

G

to

the

some

for

the

(Cineluding

a decomposition

S(t)

which

+

this, to

F we

the

will

will

append

given

this

is

a

a block

of of

from

Ca(™),

C (1)

class

then

For the

To

keep

notation

p™ the ky,

(resp.

our

superscript

the

context. is is

Cartan (resp.

functions.

sf

we

be

a

of

p-elements

an the

To

existing

we that

Cat),

example,

if pit)

irreducible

decomposition

C,(1)-conjugacy

theory

indicate

ordinary

K6™)

do

reasonable,

subgroup

matrix,

to

functions

to

For

set

want

decomposition, of

the

class

t€¥&, of

class

decomposi-

classes

much

for

introduced

this

Let

space

block

xo™)

is

all

apply

defined

CQ(™),

Ca(m))

conjugacy

is

of

the

p-regular

extend

G-conjugacy

our

parenthesized

clear

Cam),

(resp.

to

of

the

theory:

originally

notation,

of

Cott).

object

not

character of

have

subgroup

the

is

extends

entire

we

space

t=1).

introduce

the

undoubtedly

etc. denotes

class)

matrix

If

x€G

the

G-

of

x.

10-2

(10.1)

Let

a

Ed.

set

of

Proof:

iff (1p

o 1

is

)8

=

because

a

mp,

25h

ts

a

for

set

aT, }

S(m)

of

ts

of

wtf

AC

has a‘t

’

$?

where e

of

R*,

that

sums

=

particular,

0.

a

In

non-zero

e

has

a

of

wu,

k

the

definition

we

with

Since

(10.4)

applies

non-zero

projection

we

conclude

that

m',$!

k/1

with and

b® = B,

first

the

Z(R'Cg(m")).

>

ant!

m=T',

. projection

(a)

th e

Choose

then

by class

idempotents, %

show

to

WF Ug ocr’) ?

3

"yx

elements

want

we

is spanned

de Se ulep ult, AT

determined then

0,

+

ota’* .o!

an

recall

that

c(nt) =f , and quantity

onto

the

%

uCe,) and on

summand is

tells Cr’

Alt T

wm =m’.

)*

a

sum

us By

10-10

More

precisely,

if

king

where,

by

algebras,

(10.1) we

a

£9

56)

_=

definition

of

the

(1) Cult *

ES

We

have

hence

i

“pin.

ES

Cn)

then

Ko(1m)

U £

= pt.

In

(1)

=

2¢T) Kip

for

terms

elements

4g?

_=

all

f

can

of

write

group

td

o(t)*

;

dno

Ba

yields

&

2, Any 9' fag

*

CuCeg u(t

*

4))

_

=

(1)

the

left-hand

is

ulep dtr

(7 )* gs

side

of

*

CuCET gt)

(7)

*

the

above

- 2a etry g (10.4)(€a)

(tr) (1).

pe€Es

T 99

_

Using

we

have

(1) CuCki “ 4)?

the

€ S(w)

=

Colm)

ES

Now

Kno

equation

10-11

#e

because

is

we

get

(9.6)

a

* oC m)*

now

(10.4)(b)

the

that

a

form

f (1)* Ts’

0 g6M* Tod

are

To!

linearly

if

B# b°

se

Be

pe

independent,

we

conclude

w,o' = Sg" °

We

are

now

of

the

argument

Iizuka

Aang

that

rh a Since

eo m* L

implies

fact,

In

idempotents.

o

by mo

Je

But

block

of

sum

[8

ready

for

the

are

we

second

main

theorem, is

here

following

due

The to

],

(10.6).

(Brauer's tEd,

second b

te

a

main bloek

theorem.)

Suppose

of

and

Co(m),

¢€&

b,

Then:

a)

epta ed

=

for

ae b)

ag

=

0

unless

if

B

u

from

ule)

X € b®

b »

G

of

for

alt

blocks

B

Ge

att

X €

Irr(G).

10-12

Proof:

first tion

We

step of

is

the

{f_, ,,} T od

will

to

expand

{f_,

are

a

prove

>

epf

o'}

»

basis

a)

and

for

as

Ty>

Which

b)

is

together,

an

R-linear

possible

since

The

combinathe

R.

aA

3 DO

mJ]

™1

+

=

&

ww

ow

ae

"

>

From

=e

12

G

=_

TES

XC1)

y

[S]

nef

1

ayy)

b€Br(C(1)) 49

Tere, 4 we

XC)

5

=

wy (Er

a” WLR,

beBr(C(m)) “x¢

°xe *T5¢

yar,

obtain

fis

e,

ex

LBC p {Vin (Xi, 1

Je

op ;DR

Ps

TPs

10-13

1

XCLY

Kp,

yoos

oCo Dept l fay 41 o' Cozde,

xerrfeseay aH

ie] and

[xCmpz ey

1

rare

d

a 4 | regtem

2,

p

(see

Baty Vo? gle

hence

fro

=

Therefore,

(10.7)

For

fixed

tw,

define

(7m) ’

i,

Vo

0)

: XEB

T xv

d

TT xo!

pre

(6,.12))

10-14

for

all

6,

v

©

Br(C(m)),

Eff

We

know

for

that

A.

2

since

roots

of

x°%

unity

integer)

and

generalized is

a

rational

number

we

whose

conclude By

which

o>sVv

€

=

R

Ley

of ay

because

.

the

f

are

Tso

a

basis

Put

h

then

o&p?

then

we

€

B_

y

a™

yép

for

over

XV XO”

by

(ay y”

decomposition integer.

write

hy

3

(7.10)

y

og

(where

b

of is

the

pth

any

=

an by the uniqueness of x ¢ numbers, it follows that h g

Moreover

is

is

we

have

in

an

L

automorphism

Vs

denominator

that

qt

every

Q

since

definition

can

_

vid

a

in

power

fact

obvious

= urs™

ve

,

is

3

of

p

a

and

a rational

matrix

rational

therefore

integer.

notation

as

10-15

We

are

interested

corresponding which

lie

to

in

submatrices because

Brauer

blocks

of

p6™)

in

L, has

for

Lao.

Hp

block

form

we

_

rs”

which

be

definite Hence,

consists =

B.

so

det

The

det Le

=

Help

of

and

can

and

RCC ee in

obvious

notation,

det

However,

if

we

~ hy g!

B

we

conclude

H,

2

B

det

C

that

(nm)

B

put

=_

apexT

of B.

H

Colm)

Denote

these

Then

write

rem rp

we

positive

Since

B

=

and

respectively.

(10.5) is

B

b®&

Hp

By H

¥

blocks

matrices

>i1,

L

Cm)

those

La 20.

of v,

which

by

Lp

where

entries

characters

b

H

the

dy T 6!

are

have

of

p6™

positive

Lp

definite.

=I

for semi-

(mod

p).

10-16

and

Hp

with

b®

=

Ch,

=

6!)

B,

for

then

v, 9’

(10.3) Cm)

_

Ce

The is

~

matrix

positive

~

Hp

definite.

with

equality

[To diagonal but

by the

the

see with first

expanding fact

that

Since

conclude

we

that

+

~

Hp

+

H

iff

Hp

=

this,

we

may

B

) #

zero.

are

determinant first

already

H,

=

0

=

~

hy

det

H

while imply

Hp that

B

assume

In

that

result

then

follows

the

first

row,

using

minor

of

reverse

d al

is

then

a"x6!

Hp

is

inequality,

particular,

=

Hp

and

The

along

the

that

entries,

principal

have

0.

semi-definite

first

of

the

C(fq)

0.

real

the

of

conditionsalone

non-negative these

b

°

positive These

det(H,

blocks

yields

Hp

is

in

for

2

96 €b,

all

positive.]

we

10-17

and

therefore

proof

of

over

@'

has

a),

can

block

only

extended

to

bv

all =

B,

enf,

By

29

we

note

be

taken

form,

non-zero

are G

assertion

if

B

all

zero. we

but

of

(10.6)

that

in

equation

only

then

=

over

by

part

v®,

in

x € Irr(G) Thus,

b)

6 =

$

0

proved.

To complete

(10.7),

€ b _ the

which

2

the b G

if

the

rot

over

it

can

yx

B,

is

be

missing #

the

sum

because

sum

case

5 yS™

wes y

terms

while

we

dv iay |

if

f.

[6'eb 6',% xeIrr(G) XY Xd |

Tov

have

47

es

eter (gy xrxe"

(1) °mat! v.9!

therefore,

(1)

orl],

The

second

consequences 6

9@'

because

epf.

3W

and

is

get

»=)

(10.3)

b)

anda

ry

(1)

£

=

Lc v 9/9! ‘ofa.

main

some

p-element

of

theorem

which a,

has

we

define

now

a

f

Typ

number

state,

.

of

For

ao

interesting

a

class

function

10-18

6Cg) 6g)

Let

06

be

tT Ef (6,2),

Proof: Each

class

function

Recall

and

A,

€

s(n)

if

g

€

s(n)

aelass

funetion

let

bea

B

on

block.

that

6,

6

= yey OP OX

defines

which

we

will

naturally

also

call

rule

fe

For

X

an

=

irreducible

XCeyr)

G

tLet

fhen

(6p:

=

function

on

g

= 0

(10.8)

if

k

character,

C1) Tap

ie

X

we

have

~ 2 XGROX BOX (k))

ITY

(1)8y

by a

6,

definition, linear

by

the

10-19

Since

6 = }(®8,xX)x

,

x

we get

6Ce,)

Hence,

we

from

the

=

(8,x)x(1).

identity

obtain

-

|k] xCx,,26

xX

Ce)

[1x Cx,) C8 4x) (8.x) xk)

and

therefore,

o(ke,) The A

second admits

main both

=

J o(Re,)

XeB

theorem direct

=

x

says sum

J (0,x)x(k)

XeB

that

the

basis

decompositions

= 0,(k). {f.

9

oy

of

@A_s,

}

Ae

"

>

> "

™

10-20

®

B

In

particular,

it

follows

E Cena)

where

EL

conclude

is

the

B

e

that

=

en(E (ad)

projection

of

A _

for

onto

all

A.

aeéA,

We

that

(@cE_), Ca)

=

QoE Cea)

o(epE (a)) 8p°

for

We

next

generalize

(10.9)

If then

EA

all

@)

a€EaA,

oO

(7.3)

x,y

are

in

different

p-sections

of

G

10-21

y

x€x)XTy)

X€&B

Proof:

Define

a

class

e(x)

By

the

class Hence

orthogonality

of 6,

y,

=

=

function

f}

relations,

-=

(65

(10.10)

Let

m,n’

@

if

(02),

=

ay,

via

vanishes

off

y € s(m)

so

€ f

9

xCVTx Cx),

xEIrr (G)

in particular

-=

0,

i vanishes

8,

,

then

and

let

B

the

6 = 6T off

be

Ale

a

O

block.

Then

—

7

yen

XOX

Moreover

Proof:

The

first

and

(10.6).

The

second

can

perhaps

in

two

ways.

be

seen

'

1

d '

Bk

’

B

=

6

=

) ned

Ts 7

assertion

more

is

a

c

(1) $ 9 b ‘

(tT)

is

by

immediate

B

=

eB

e

X

of

computing

From

e,Z(FG)

?

) R b b°=B

consequence

easily

”

.

Z(FG)

from the

dim

(10.3)

first,

but

(e,2(FG))

°

10-22

we

get

= He.

dimCe,Z(FG))

Z(FG)

the

(10.6), result

and

we

also

have

rev | “4 @F

epZ(FG)

by

However,

Legh

since

f

“B

the

fo

3

T 5

$

are

linearly

independent,

follows.

Finally,

we

get

an

interesting

result

on

defect

groups.

(10.11)

Suppose untess

Proof:

with

b®° =B

exist

then

By

7

(9.6),

iff Xl scm)




then

Ay Oo _= ,0 Ay

-

p-rational,

a5

e685GL =

Ay

%

with

A,

= A,

+

=

some

_

eG,

-

Ay

+

e@,.

4,

and

and

=

sign

e

e

exists

O;

©

A.

Since

the

A;

that

®@; ©

B

that

o)

B.

€

are

s

For

i=.

But

i,

from

follows

that

therefore We

8, have

irreducible for it

any follows

i,

©

B

the

it

the

then

-

by

(11.2)

(b).

by

0.

e(6,

for

obtained

in

Oo.

=

=-_=

Oo,

1

>

then

are

by

distinct.

To

show

(7.10)

to

show

equation

#0

characters

Let

suffices

d;(0,)

therefore

A;.

-

(l

>

TX

composition

permutations

previous

of

result

tx.

part

uniserial.

therefore

Y*

get

= Y/tY

it

is

factors

of

each

has

is

we

TU,

Z

same

first

and

count

conclude U

the

xX.

Y

the

so

> tX

of

then

affords

Y_

determined

transformation

(11.15),

a dimension

We

have

the

in

and

that

= U.

R, G-module

with

then

X

proof,

=

Y as

as

the

Using

RT

implies TU

be

identify

character

of

Y)

= U/tU

obvious

xX*

= U®.

that

and

U*

other.

the

Oo:

following

interesting

consequence:

Q3.16)

Suppose that real

Proof:

NCgz)

=

i

for X;

some

ts

Brauer

For

y-t¢g)

tation

a

i

(0

fhen

vertex"

X,

is

real

it

is

iff

b

(9.16) Ea:

and

immediate is

B= 8

graph

(BB)

ecanonteal

more,

then,

the

ts

planar followtng

of

(9.2)

iff

certainly

56

Sg we

we

notation.

and

to

a

there

otherwise

choice

from

above

then

N-conjugate

summarize,

fhe

1,

real,

proper

that

To

a

by

the

(9.14) and

jis

N-conju-

have

tree

which

embedding.

has

Further-

conditions

are

equtvalent.

a) b)

B xX

ts

real,

ts

0

real,

c)

Bo

ts

and

if

these

then

complex

morphism fixed

of

points

polygon,

N-ceonjugate condittons conjugation

s9(B) are

to are

satisfied,

induces

of order a

ee

connected

2.

an

auto-

whose open O

§12.

More

In on

on

this

defect

third The

main

is

that

of

two

Sylow

is

due

to

(b,P) a

such

defect

third

main is

of

PC, (P)

PC, (P) |

a

block

strong

of is

The

results

known as Brauer's

follows Green

[ 7]

always

an

proof

Passman

[9].

which

intersection

we

give

for

this

theorem

is

essentially

a converse

object

of

interest

a pair

with

(mod

pair.

p),

(b,P)

ts =

PCg(P)

a

P.

CHC

G

then from

ts

a

strong

§9

and

H =

PiC,CP).

b We

call

is call

block block

then

(b,P)

that

= b}

NG(P),,

P, = 8(b"),

P.

we

Furthermore,

and

and

in addition,

= {x © NQ(P)|b™ (b,P)

is

group

If,

Recall

Suppose

§ (b®)

of

defect

pair.

Z# 0

Ng(P), (12.1)

usually

a p-subgroup

a block

INQ(P),:

further

Scott,

P

a pair

is

result group

main

The

two

approach

p-subgroups.

where

block

our

a useful

a

prove

first

--

Leonard

(9.14).

we

The

theorem

states

Thr

Groups.

section,

groups.

second

to

Defect

. pair.

pair

fhen iff

if

|H:PCQ(P)|

= py

|P,: P| = p

Proof:

for

any

ecb?) Hy

We

subgroup

= Pp

d=

Xy

of

o5

Let

Eb.

X,

If

some

(b,P)

v([H:H,|) we

= Pp. and

(9.6).

Put let

Then

is

Ny,

=

an

irreducible

for

=e

Hp

Hy s

se

h

(see

X(1)

=

a

strong

because

in

1

Caza t

e

be

defined

By

N=

(9.8),

NQ(P),

ct

be

the

canon-

N.

-

Now

suppose

and let x e@b",

integer

is

pi

by

hy shoo--+ shy

representatives

for

that

PC, (P)

b.

has

0

2D

d(b),

H, CH ON,

(9.10),

note

s(bN*P))

character

that

H

aff

= PC,(P),

ical

first

By (9,13) and constitutent

a

set

H,

of

coset

then

1

(3.5))

.

In

particular,

est, (1).

Hy

block 1

CH

pair,

then

v(s)

so if we take

D ’

=

H = N,

get

v(x1))

v(e)

+

vCs)

> v(|N:Hy])

+

vz, (1))

+ v(Hy)

vC{N]) = vp.

- v¢(p4)

12-3

This

implies

by

(9.3)

we

by

(9.8).

Now X

be

by

apy

P

§ (b°)

| H:H,|

which

pe,

8 (by

thus

constitutent

a block

(€9.12)/(9.13),

and

HCN,

irreducible

X

Since

b,

exercise

let

so

3-6,

e= 1

and

vQXCL)) = v(/H,[) = v(pt4y, We

conclude

and

let

Since

Cg

x

a(blly

bea

defect

is

and

PS €

+

1.

Let

class

for

pi

kC

Hy.

By

(e)se. cH =e).

of

implies

Maa

that

Syl, (C,(x)).

conclude

d(b)

Py

= § (pb!)

with

xێk.

(9,13)

H owever,

* eH

and

C Hy

by

thus

definition

(9.7)

>

p-regular,

“80 Mi HS SP

(9.7)

by

k

that

that

| P, P|

Finally, can

certainly

so

8 (b&)

7

if find

P,

0°

p:

Since

PE¢€

Syl

Since

we

=

Pp

| N.:Hol such

Cy have

and is

a

0

a

b

(x)).

dtp!)

that not

subgroup

6k)

#7

0,

By definition, >

d(b)

H = PjHy

prime H.

to Then

p,

we

= P,C,(P)-. we & (by

>

P

CO

12-4

We

gives

see,

then,

a bijection

defect

group

pairs

(b,P).

bp?

when

use

the

pair

and It

(b,P)

way

and

the

block

not

result

pair

to

1

=

a) PCP]

ce)

In group

this

by

1°G

situation,

Namely,

CNG(P),

P.C.(P)

block

what

happens

this

in we

(b,P)

and

case,

another

(b,P)

extends

b) P; 9 PC,(P) (12.2)

construct

b®,

Cb, »P,)

In

with

strong

consider

"extends"

b&

G

of

strong. to

correspondence

of

N(P)-orbits

Which

satisfies

Brauer

blocks

remains

is

previous

the

between

P

(by »P,)

that

we block

a reasonable shall

say

if

|P,:P|

= Pp»

= P, =

we

b

P_C,(P)

1°G

have

the

diagram:

P,C(P)

CCP, )

to

following

sub-

that

12-5

(12.3)

Every

block

block

pair

pair or

Moreover,

can

if

block

patrs

then

(12.2)

is

either

be

extended.

(b,P)

and

(b, »P,)

satisfying c)

a strong

(12.2)

holds

iff

(resp.

C4?

ts

of

(resp.

are

(a)/(b),

C(P,) where

ft

character

then

Proof:

If

there

exists

H:PC,(P)| G there

is

a

(12.1),

1 Hy

pair

of

May

assume

prove.

Hy

all

blocks that

Put

Be=bD

(9.6)/(9.13),

all

p-regular E

=

Z(RHp)

.

of

Let

H

we

(7.5).

with

be

=

a block Put

and

elements py

is

PC(P), b.

Hy,

s #1

strong

pH,

By

(b,P).

(a)/(b),

of

not

then by (9.14) with

(b,P)

(12.2)

is

H CNP),

Cb, »Py)

that

eanonical

b,).

(b,P)

P, 1 = 6(b¥),

extends

character

set

ep

block

suppose

canonical

By

Put

P,C(P,);

pair

subgroup

= p.

satisfies =

block

a

(bj »P,)

Now P

the

b

the

let

with or

By

defect

(9.7),

YH

ep,

of It

=

H now

and

%

be

the

e,. lie

group is P

0

In in

follows

that

= PCCP),

Dy sbo sere sd,

there

get

H Gg

else %

pair

>

be

Pi:

We

nothing

(en)

ep)

fact, Hy» from

the

we

=

-

to % en:

since have (7.8)

12-6

that

p

and

Ho

=

Let

&p:

be

Then

KP

k=

let

k 0

a

p-regular

class

(9.16)

using

an(k) = a,(k,) = TET

= tEeT

|

have

Levee eeeeeeh

= TH Teo“ PEO

(12.4)

we

of

& k.

x,

, where

by (L) bg OAD

! Now (1

let

(1

(12.4)