Philip Hall Lecture Notes on Group Theory – Part I

Philip Hall Lecture Notes on Group Theory – Part I

Hall, Philip, 1904-1982

This material is a set of unpublished, hand-written lecture notes composed by Philip Hall (1904-1982). They were written in the 1960s at the University of Cambridge, where Hall was a professor at King’s College.

Part I

§1 The laws of group theory. The symmetric groups (pp. 1-5)

Hall, Philip, 1904-1982

The pages belong to Edward Spitznagel, Washington University Professor of Mathematics, and are xerographic reproductions made from originals held by (and with permission of) John G. Thompson, a contemporary and correspondent of Hall.

1951-1967

Washington University asserts no copyright over digital reproductions of this material, and cannot give permission to use the materials or otherwise grant any exclusive rights of the copyright holder. It is your responsibility to assure that your use of the materials is permitted under applicable law.

LECTURE NOTES ON GROUP THEORY

B
Y
PHILIP

HALL

I .- ~-£.~~;- ~1-,,,-~~t('4~.

~ ~ '1 ~ ~.

] 1. ·

) .

~ ~ oj

~

f.;.;ja,

e-

~f.;.N ;
~~

sd=

be.J ,:..-~

t....J;

tHi.~ILJ . ,:... fL. s~

1-

,.e.Jr~ ~J v.e.&cu

e~u;l;

ft._

'1 ·
for .lt.Xo.-.r~ , ItA.. ~s '1-

rij~ ~

F

h;- -So.:;

V;

~:~

1-

/4

w-

~s

7L... Sp,..,dn;c.. pu;s .

~ ~

of.

.

~~

.)(.

,,ce

~

'4 . wUt

1 X kk
X>= }

7 --.
~r~ ·

~~~
~ ,(

LJ /k,p~

ftc. ~

( ~J. ~ ~ !.e. ~) ~

kh X

~:t

fr

j

f~ ~ lf)
~

X

,.;.,_

r~,J-~

o( ·

~

~~~ i:' ) /4 ~Wt-«. 1-

:r•J..- J=

h~

--r4
"it~u

it.

t-~ 6

u
-

'

:xo< -=

~

'1

:..,

It,..

,

u

~

/:2.

~ k1:

1tL s-p:~~'- ~ . ~

X~

p

4f<.

JX I .

If

«tr~"'..k-.t

tL:.

·:.

~4J..

.

1-· X .'u Jc..~ /,)

2:lX) tM..J

If 1~1 =- '"" .. u; ~;i;, > ~

k ~ .;.. 2. (X) • All 7/..c.~ ~~

6j
x ( o(f)

'1
cf X ~

/k 1'&.Ju.ik

OJ<-v.Jvr~

X.

.

:>~

Let o< ~

•'

,._J l!l) ,

-JJ.- ~ ~~~

tf.

.,._"~

'~ 14 ~ IU .

X. ~ k..~ b;~

fwo f~ (!)

.r~ .f.t (:X.

fr
"(-:.. ~
.

X) . = ~ !

e~J

~

~b

fo-r

X.o(

., ;·

..J ~ 11u.k fL...
~u.~~.

~

rrll.:i..r .

~ .s.d

~

~

.x..o< ~'~o< -

fw-.-

'1 .

"'~

~.

.:J(.

o{

x

-~

'~ ·t

.

(x.~x)

~ fwo

e~H.~

_
c..-.J lfi)

;

JL.~r-~ wJt. ~<~

\ .. , ~.-r'i'

b:l ~~trp.c:J-~,

k.~.J&;Js -~ . ,:.,

d .st
~ · IJ..a.C I~ ~'P ~ c( i.. (helL- / ; , - ~FNL
skf~ - ~ ttf /k., ~lA- j = l<.oC . .:... ~~) u,

t.-

F~ ­

.

A

·

~ ~~do~c..

~ ~...L.r s..L:J.

w-u J...r\~~

1

~~

,, . lk

liL ;e~J~.s

-:: (xo<)

71~

lk

~ lk, ~;.:.r "J

...

p.

~~~ ' ~t .a<

O(p

(x (:X) .
:>

~

p ;

sd 2 tX) · .. .

~~J /• ;,[; ;_.

~0

e~ ! ~ x.<>t<~

~

lx

~L:_J~ '
l~~~ ~~ .
~ fw.o ~ sa.l:.f:J ~ ~ l..&AK~ ~
I . (o<(?>) ¥ = 0( Cfr) -' lk ~~ ~ 1- w...Ji;:rfi.cAI~j

li)

C)..

ir . (oC1J -I-=I!L . (_«. ot..- )f

<X.

"'-J..

jl

-=

1

N~ !U

;

J~ f;mr~l4-cf

f

=

(eXeC') ~

.de_~ o;- X ~ ·Ji,~ .

a..n~ "vrro...~

1- X .
NtJ-W ld: G- 4

JJ..

0( ,

~

k~

p a-; r
~d ~

~lHe."'j

~f.,(.~~

~<.r:d

D(

l o(

fo-r

aJJ.. . o( ,

»J.:u.J.c

wLJ.
,p w a

oe

~~

AI'C-

o{

f
a

l -oc.)) -+

p

~

~

1

c).

k~ ~ L<! . . . ~
ft.. clvri-~
~ ~~

JL., ~ I

WkJ

;J

71:.

4-.J 7!L

11

.

W

~

i""¥ .

Cl.

lt.c., ~"'<J-~

I1..H.

6-nU

a.w.c.J ~

Jlvz_

~ ~.t 1U..~4 lb., ~t 4f}~"T~ .
Ftl'f" e.x-.-.,1.._ ~ lfv_ ut ~ '1 a.il
I

11

a-c\

J"'~" ~

~ , "'"'J

fk ~

'

j

-=

p~ f

+ ( o( -1-

~ 61. . . lJ.t, e.J(

. Tk J.iA t~~

~~

es

(- 0())

r

t'U4

~

~, ~ c.a.L~ ~

/,J~ o._»;l-~ ~

w~ ztx)

~o-tJ~.
s~ ~:~

·.:._ ~J.. " ~~~~

sJ

~..

G-

a.-r (f+y)

'~~ ~

~

~

-j-

p

~

=

(- oc.) -::.

-

Itt...

.:-it~~ .

(«.+f)+ o

. ".!JI-r .

a-.J

t.-tf&

p~

,,1\.L,d~··

'

11::+.

0..

/L.u., ~ ,

?lht-

.

~ G- ;,

«.-I )

}>~

I-r.

4llr .
A

·

~;~ .-f X

'"'

O..n:J

~

~t !~ 1- ~~ .

t.. J

~.,.u11Jti--.

f<-~4~

B) .
>~ If
lv,._\H!_

t '

ic4J:Ji

is IJ...,

o{ o(-l

"J-

If.... ~

t.

~ ~

f.Juz_ 6{

1- ~~~, k".J:

14 :J~- ~ x,

7L a-) :JI£.

~

--·
'~ ' 1\.ft.:J ~ (Y ._, ltc., -(_~~ 11-l ~ ~~~
1- 1 ~ G . It . io CJc1LJ - ~ '-V\.;,t ~ 1- G- ~J

I· I

J~

j_.

7Lz, ~'J:: ~

~ lie., i~ ~l~ 1 X . f,__
~ ~~ ~ l</u.- <Q. .) ~
~ 14 ~~rfl 1 ~ ·~
t~ tt:.. 4-4-~ ~ I~ ~ "-.tal~ ~ o;J 1\~ i • 7L- /

1-

2.(.X)

.sr-Ju.
1

"I

+ - ~)

-r-

-==-0

( ?'~~~

JMf~ck

~,._.J

~
.r~

\.f:s r~

~

~ tk ~ ~ ~

1 (;.

~~~~ -~- t~

I~

F(f(" e.:A~&._

s,' 'f~

~ ~

>

~~

~ ~~

<Q .

~ ~~

.e .... ~ /·.2.. .

1

sf

dL

J

••• ,

~

'"

~ G- )

4..

~ \4"'~

~ ~ ~

a.J

Jl.0{-

"'- . /tu.. ~~

~~t.:.c.J-~ "". ~ ttuC.
~

,

1\

= ~ ,

.,;2.. cctt:z.)1J)t~

,

fk"

~

-uc.

Lf,!.z..)c.r,)I~)

>

~

c~~wL.f~

1

(s, c.rz.!.J))J,.,

r,(Cf~I.,1.r~)

-'J'k.. ~ law I -4!~ Ita£ ,ell f:v< jW<. I~
6.~ ~r.
?kf.NO • ,:._ ~ ;~J:. 1- M.~ K" ..... ~ .sd,_~.J.;, / "/- "'~ ~ b c/c.Lfs ~ k JAA ~J wUt :
~ ' IU ~ ~ ~ f-.d-"""' ;,. ~ ~/'~J. /""
jewtrd 'f7 ::/=?'f .
If ck- .s~ ~~ IW !, ~?'I., ~ ~ ~&.~
a,.J 7 ~ J...AJ & ~~ut. .
~~ ~ ~d ~ p~ i ~~ .
", (s2. LI3f¥)) .

lr

<~~

~"-m.

~

v~~ sf-<-~ c.f..u

~

$ 0 2. - 2./ ,

J

c.../k.J Ak.l:.a....

[Ju., ~ ~~ ' w(;_~

~ N,H. ~

'r-o-«f?I

~ ~ ;__ oJJ;.f~ n.ot"J~

~~ ~~-~ ;~
~
~

'
fL_

~

~

"A. =/;)2,3>

Le.,._
I ~

I

I(

0(

<){ J\ -:::

c< a-J

1-

tl"'

~,.,. eJ «

1

A.

~

~ ~~~ ~

D.H-

'
-~-1

i _,

o(o_

o{

=

_,.
o(

_,

.0(

. • •

F..- .J.L "', " .:.. ~ , we /.....-.
o( tt o( "' ;
(o<. tf\) rt '= o{ "1\1'\. •

I· 3 .
c( tJt\ ,._ 1\ =

p ~~

J

~ d,.

~·

,

(o(p)'-'-. o(wtp~.

/
\

]) ) . ---rk,
r--

s~ ~J-~

,..<.M(:

~ I· '+- .

f

. ~-4-br~ r (o<.)

~ G-

~ (c{)

I

I~

Let- .~

~ ~Jai-~ 1

a.-~

£"' )

I'

~

~-

st~~

p~

c<

~

~

6- .

=

o() ()(_,

t;r

fU

-=~ f3

s («ct') 'd-

>

flu_

~~
~

~0

i.D

[3)

.

~

_ll:/f

frq_~l.:f~

~H 'i G-.

11ut.. ~

w

~,~

oe -::

.-fk_

~~

.l(o<.)

r-'.r

~

L.e.~-"'~

~f'4

~oc -=

.

=t

frr

lt.k > ~

p

l..tU ~

F,r

fol ._ f3

~'4- -:m:. ;

~-'1r- .s&t.J-~

A..

..:-;~

~ (par'~"' =

I

poe'

ttJ,

p (o<.- o(.)
1

= (.~C') -I l :f ::n:. •

"'-"'-lif'H

If... -~~

"/- G-- ""' lk »'is/.r

1 ~h /~ ~ "' (!,. eM ltL
/X~~ 1- G- ~ tc «'41·,4~ for ~ o( ~ f

..Lr.vr

wL:cJ,

~

&.,

u.'"tr-'~ p.14-

s ~~ '} .... ~'f
-l{o<)

"(;

!(f) ~~

"-o

1

G- •

ot~~

"r,..,.,- tJ~" -u
~

1 c;..

4.

.,..(1) = -l(J.)

~.....:Ju.

~~Q.ri.tt.l' .

~ · I~ ~

fiAJ:;

J

y. o( +-, ,

FIH"" ~Q.-t,!.._ )

-=- 1 .

'1 ~ ~ ;~J4 if 3~ ~~

l·s-

~

If G .:. Mttl;.,_, ..f(oc)-= ,..{o{) fH' 4ft ~~G-.
,.. (") .._J ito{) c.vUt '!( f: G- ~ c..JL.J 1/v:..

~4~

~)

1

7 = f3

~

-r

0(

~(f) -l.(o<.)

-=;

~~

o(

r-(GC {l) = .,..(o<) r-(f)

l(o<.)

...

~~

f(O(

f

X ~

If... ~~

c<

"i-#

2.(G--)

€-

~ 7!L , en;.. CL

7k. ~~

•

~J

c<

r(O()

AtcJ,

Z (.G-).

, r(oc) -= t-{p) ~ 1 «-= p.
L-..Jc. ~ r~J~ fro-- ~J.. ~ s1d41.

1t.. ~~

6- ,

"'"(o<.) '-'

~ G-

lU)

G-) ,

f

1'-(oe) f

Tki,o ~4~ .i (;, . ~·C'.

I .)

c(

F~

k~

cr

·

i.~.

& _,

= -f'(e<)- 1 .

~

0(.

&.

de.~ -f · ~ ~ G- • T~,., 1k

41\

~ ":~

t ~ .r

:

k

~ ~ ~

fn,-..

t~,I..l.·-·s~)-'-==

r:',·'f:'s:'.

I

I

foe-=!

E)
/'"""

/""v-e rU4

I
L
~

;~J;

~ L.(L ~ ~ '" ,~ ~

1 "-.

A :> B > X I> X~ , ...
A_, 14 14 sJ 1 ,JL

~

AB

w

14 s~

"~ ~)

Jk.

,:....

1

11...

~ I_,

ot-1

~

~

~

.IZ.x~J

I·S

o.1:

-

B::~

l
J~

~;~ ~ .r~ ~

~ I~

I· 'r

we.

Le-.-a... I· t

L..ve-

o(

~ ~-1

~~.

~o

!

0"'\c.c..

a

lX1 X

4 .,.

~ ~o.rt.....:vJ

G-)

(._o< t-

S~ it ;,

0(

4.:k

~ A .;

A.

;..

~ Of~
_,

_,

s~ :

(M

-1 _,

'

X,.) -=K" ···)(~X,.
CAI)U.

- ~ f-<~~

~,__1-J,h._

:»

1- G-

~ ~

...,l~

/A-'1= /AI.

~~ . .
fqr

oJl J~ A _, 8

I A t l -= I A I .) ' I '1

Oi ~b

~

.t ~ G- wLd. . c0e- -bt.J>~~ ( ~ o.l: ~ t
{o-t- S-= otf ..:JJ: oCfA, p~ B.

:JI

~~

14

"'-

alL ~

(A B ) C. = A ( Bc.) ;, (_A . )'= A ;
III

I

~ G--.

~~

_,

B~ ~

for ~r/,:J-t""fl4.7

B

/AB} ~ IAI·IB/.

/.

J

= ' I8 I.

,.,.J . .J.l -dt.-«,J;

f

>

? "/-

A.

~4