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Philip Hall Lecture Notes on Group Theory – Part IX

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Title

Philip Hall Lecture Notes on Group Theory – Part IX

Subject

Hall, Philip, 1904-1982

Description

This material is a set of unpublished, hand-written lecture notes composed by Philip Hall (1904-1982).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 IX

§9 Frattini subroups. Split extensions. Soluble groups. S-subgroups (pp. 70-80)

Creator

Hall, Philip, 1904-1982

Source

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.

Date

1951-1967

Rights

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.

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~ · ~cL.

-l~(H

A.( 7 j) ~ A.(?):r A.()")

~ ~

We

ftr ~ ( i )

7" tf

we. ~

(3 ::;

f3r

pE

K.

?k~

Cl>)

ltk

k(j)

for

n

7T k(7) -==-

o(-.:.

-f.

?t;H

.

So

/U

'! ~ H ~

,J!

(_7 > 1 E- H ) .
-'c71) -= o(r kts)7l ~

t u)
J\=

:1>

~

.f.?l = 1..

~ ~£3 =

f-;.4 cAtrJic:f,;_
A.6W

~~ ~); or

M

:r k(:r)

=:

/I( I
a....l

JH/.
l

~J tY£ff4~
Jo

C(

~

H, =- H P.

~~ (!/) .

~ikr .r~ ~ ~ ?•'r.

~ ? · '~-' . L«: r ~~ A. ~ ? a-t-~~-.4 1- ~ of ,.,.J,.r
f~ /$ /G-J .
If ~~ G ~ I i- .r~ ~ IL.k- for ~ pn:-c.f:.
l
~ ~v~o._,.k Slhto..C.. {,i.w ;-.r~ 'f G-.

lJct. "tf'rJ. 1·"Jj~ ; -.s~

1' 4- li J

c:· ~.,. l!fJ

>
J

"/

(fi >

G

~ liA.

a-.J

srtt

~t.~

N-= NrG- (S)

N

eo..::t~ . ~ a. .s~ fj

r;

~

~CAJ~ -1:s

s~ ~-~L f- s~ "'I

c; _

r t...re-- .

>

rG- .

!G.~ Ntr =

If S '- ~

rG-

"-J

$•

La-

~k...JA.~ l:s Nn G-. B;r

!Ie-ee_

r

~0

,

~ · .:...~~

A H

c .. Lb~

~

lr'r

'J- (;.

~~~vi.- f!-d-rr.1

V
"CY'- Ms.

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alL ciiJ::u.. car-~J

a.e

q<"

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CV£

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ri"N ?

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e.Jkr ~ ; -~J fov" •er-e- f f. 1;1"

a../)_

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J.u_ YA'-r"t.r.

-c::r -s~ -::=:;> -Q-~

~~

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fk_

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qJf_

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~ TJ-J~

G-

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~ocJ.. .~ itA,
"-f"";J<k fcl,..,., ~ /,._ >Cj'~ by t~/tct.-.

~ -r:s- ~J

fX"

~ ~

/_~

"'f::J

'f

;J t.a,. ' ~Nz.. J..H-

G- ~ Tb-S~ "f ~J



·f

if ~ "- ~~ ~ p~ a~ tdL -e.;Jbu- p-~.r ~ s~
f €- 1:1 d ek ~ 1-~.r.
.r~.s J ~ ~.r ...-.J J<J~
1- TS- -Uf"",_~ ~./ tV-e crir-~~ • s~~ ~ -r:.r-J~.
L~r-.A.

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G-

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7·£t

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Fu.:st. /,]; M be.
d ~CL.
If s "' a..
~~

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""'-"1

~J

~

A.

or

ftc,

G-:f=

Le

i.

/..eJ:- M

A ..

S'f-.!~

H . ~~/; J,:J 14 ,;,._~cf~
J"e:JJ.l: ~ 4 011 ~ {«" 5 . J.l.e.,..u,_

~
If '/'-=- U rr vt , IL.. rC4Ji- ~

J;4-,:,.~

SJ.t -s~,

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AUu-.«.

it .

-cr- ~ .

~"""' G-, -= . G- _,

~

4-..J ILl::. Hj/1

1

G-

~ ~-e.

~t

1' (;.

tf G-f M wJi. ')' = ~,
l.:JJ' ri-N.o,;, , w t 1-( <

ik...

TJ-J~,..Jl._-

Sf::T-S~.r ~ sr::rr_-J~.r ~J spt-.J~s.
zu- ~"--~ ~ ~'-~, tk ~ fW= G ako ~ .

.S'l:S1 -.S~ ~J s,TT' -..f~~.
f~
t\J · ~~~uNv ~ /G-/. , ~
4..

b11..

L,._

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b~

~ ~ G-

a/

E

·

1

11

~J

a.,=

NG-(S) -< (;.. _,

of M .;, ~, s1 'L-s~ , of

S <1 G- > M -=- S

G.

AoJ ~ 1Jv,.. J'"CoJ.k- ~

r

'D- M ~ n.e-~: ~IT-~ . .) :1; ~~- k tli,. iN'-~ • ~J~..c. . ="t41'".)
J
.,. ~1~ ~ ' -s~ 1 G- o.,~ ftc_ ~ulf ~ fo-... CJ·lt-. UL ... 'f-7:J
ld- T k ~ .f;jw r-s~~ '7 M - ~ b-~-= NcCrl . If G.2. < G"".. . su-z - ~ 1 G-.._ (L'J_~/; ~ ~~ GM..) ~ ... ~t- s~ 7' (
~uG-2. -= G-. ~ If . G-.z.-=- G--> ~ T <1 G- ~ T 1'1 ..._~ lk ff4u..lJ. . ~ ~
~

0

J>

, ..

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-p
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r--aJ.~

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kJk

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~

1-s-2.

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I

~

s{...._ .ca..L

.

L.d

k

G-

~t>.j STd- s~ · f S.
"
L l .::;; H fr:rr 4~ ~ ~ G- .

k

4-.J

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~ltv- ~- s ~ ~ e !_ srf-lt._
a.~::I

be..

W: L
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1

'ftJ'-

J~

>

~

H

1- G- . ~

+W"V ~- S~J 7 {;-

A."J

I

~

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p~

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~ ~

/G-1 .

~-t. 6--=F 1 .

Let M 6e. A'v..~k..-1 ~ ~ 1 G- .
~ MH/M . ~ ~ Sr:s-~ 1- G-/M
~ M /M ~ ti- 'f!r-s~. ~ t-~d~ _, L'1 6 MH frr s~ ?~ G
1k, If~ ~ l:J a.. ~J ~~~ 'tj- M H < G , ~~ H ~ -S~ -~ 1- MH.
So ~ ~J ~-.c... MH-= G- • ~ GlM .;. .._
cr:r- ~ .
If M .;, ... ~- ~ ;) /4.,._ H -c. G- "--a ~t«, ~ ~~ lcf.
k- r~
If M ~ "'..t A. "CY- ~ , -J: ~ A. 7\.,.,. ,J. s"ib s~
o-...

.

1

I -

t;, , .s~UI-

G- . ~ ~-s.er~-

F~.)

H n M-= 1.. ~ H ~ ~{M -

~~

~ k.~p~ C.:. A-t f.e....ot
/k, ~..s M ~ H ~ .r,t.JC...
~
4£,<) L ("\ M '::::.1 o...J 'f G,::::: LM ) 44. L, -.::. H A G-, > ~ L ~J L, -...e
(hUZ..

st;:r _ .$~.{ cr
~'i

t-M

E~ L ~ M "' s.,u.& .

&, •

1'4-(Ji )

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7L

J,~ ~

~

.r~

,,h-e., ; ' Kfk..c.d

':1

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~ -u-LJ-~v:~ ~ '1 pn~ 1

G..

w..o.JJ:-

~

~- 'j·S

IY -fvr-

l~ ~ ~-

a-d

F(/'(' .r-We, ~J, ~ ~ "'~ .tt.h.
~~

fle.cc.s,_ L, =

L~E; (;..

k .t~ ~) ld

G ~~ ~-.~~~.
(f/ J ~J in, ~- S~s rf

""Qr

'.a. ..~j .t4l 1 ,n:wu..

~ .... llJ

Uii)

Bir

E~, -c;r-.s~
~ · s-2.. _,

'< f G {;,

1 G-

~

"rJ -

G- ~

~~.J._ ~ G~ ~ .;.d ~ s~ ~-s~. '1- G•

.s~

K"

-ey'-$~

...

tf..t._

~-J~

'f & ~~ ~ Jv..c.HJ.u.J o.lJ;;,_"'-~ .u ~ ~,;,....c.l ~-.f~J '1 G-_
/(- ..,.;;>.. {-~ fk.t ~ S~- ~ H ~ G- .:, t.~~ .,;. G- q...._J
t . · I o- "N) ~ o....~"'"~ ..:_ G- .

L~~

/I<

1 ::: I G-

L.c. ...r~ 1 & , ~ a-J L--It" K..
so 1 ~ L 1 ~ /lr ~ H I . /G : " 1
G

tl....J 1<.
~J

: HJ

a.~

Ji.t;M.cJ;

H a-.l /<.. ~ ~~~ .:... Gj 1~ /I<: LJ < /G ~ Hl

i</ = h

JG :

<Af~ , It;.._

o.,e.

HK< <

4..)

I G-: LJ

= ~7\ ",J

-== G- .

f~ ~

/~ ~ L/

{/}
~~)

/(

Lei: H

If I G-~ HI ._ "'- w

(!_'ii)

HK

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If

ii)

?·&

Je

If ? f H ,

t.~"-J,J;._ ,:.. G-., cu..J

o.&e

s~

(_iii)

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.

-=.

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>

cl;,.f~cr

7'1

J

c.,

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, ~ HI< < G- "-~
/t:c,.

'f

~ J< • J.k....c.L

1 "'-·

: f1 J

S~ -.k

~t.. 1

A.

H "'

/I< : L J < JG

So

}G-:L} :.. o.. -..J.,ttt,_

s..:..c..c.. (_~ .. 1\.) -= I :> IG ; L} ~
(!iJ . ~CA- I r;. ~ Lf = "'"- ~J

J,~

b\."

K..

s·r.

bJ

L ~ H ~ G-,

"- ~~ ~ ~ .

&J;

HI

'Jk)\. •

/I<: Ll-::. ~ - a-J

So

HI< --G-.

~-- j·i
H,
G-

>

~

H2.. • H3

*

P~ ·.


~

IU
Jf{,
So

'Hl,, ~~ , .., .3

~

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C41'~

,;._ r~~

. 7k...

s~ .

io

Jo

~~

tll~.s .. ,_e. H,

C4l"

. /M/ -=- J>



H, -= 1. , tk-, H :L. = H3

for

)k

~ ~

-=:::

1.¢ M

_ 1.
=/=

.s~ f~

G--

~J

~t.u<,

"-~ b; t~·

k · '""" ,~J, --~ J~ 1 11
S~ (,~. ~~)

f .

i!.>

,.,.2. .

=I

..

W<

~ ~

d'J Y· b C!_i,}
L,4 -= H, A H2.. • 1/:i..,
; L14 J = '".1. ·.
J.~et- L,2. c4-M "" S~ ~ ;-.5~ toJ )-J, c-J
M ~ Ll;z. • A{,o G ...,.. H1 H~ "-'<> .to ~lrl.":J .d.t.~-J- 1 ~ "-s /fc.

~ S EH

{n- l7
M 7 = M?

i4

~ 1\1

~ ..:. G

~ H~/ K n Hi

:.;..C-JL


3\M.,L,_

H

~

~

l1:.cJ:

1fHz.

~ . <.Y
"-... )

suL.Je,..

I

{}f K

·~

~ H2

/,... fr/K
lk..<

>

"-~u.

'""-k. "''- -= I G: H,:}.
~

"'/-

7/..~ J~ ';u e.~~~~~

L4 ~a.~

.s~

=- ( G:: KH;_

a.r:.¢ «.(.n..e
.

:>

1

J

.s~ .

~CA.

lie-u.- J<.-= {_M? > 1 EH4 }

.

fk .rw.%~s I<H.:/K.

-i

ct,e

"")'~ ~ f~

/3J .-J.v.cf~ ~ /6-/l> ~
t;.

~ .s~ .

----

---

--

.-

u=

~

~

r

1 ~ ~ ~_, ?- 8 ~i.k. ~h ~/A.,~ 1

7k, t~

~~

~
~

~

7-s

, L

If
G

~lLos1- .

G- io J~

. f~
1;-G ~ i .
~

&

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•4
}

~~

I
·

tL.,

H

G-

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1\trt .s:_,l... .
16 /G I ~ ~..iJl..

§
A.

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1 ;.,.. /'~1.

~j'!f.J;. ·

;,o

L:~ ~ f.,.,.,}~

f

~J

t,

~. .



~ ~~w.- ~

/G- 1.

lJ.c. ~ ~-c. lt.-k

&-:j:;

t

4-cl

u

1L- G. ;_.. ~ ~ ;-F7 . Let 01. k ~ ft~ t-J~
7k- 6l. -+ 1 ~J .so } 6l. -t 1 . . Let r €- } ~ ) ~ -+ 1 .

G -:::. i

/:,J' '1·4-- .

w\lL L~ r~tp~ lr

we.

>

~ ·I G- : CG- (!)I -;.

i"

> I •

/~.., ~ ~ s~ K<J G . ~

So
J,~

~

~ 1·[

~ G-jJ<.

4

fo-r

~

w-<.

J~.

}/.c_c..c. G-

~ t\.oi .{.:..,!.._

1 < K <G-.

13J

fk-o.. G- ~

c. .. .s..wt.r~ ; .

.rkU.
<1. ..~J -':; f ik L.. ""

/'/- s~
e~~
f~ : /~ ~~~ ·~ ~ t(J ?·S.

f:. .--

lJ /GJ-. f,.._
![•
~ ;, < !~ < ·--- <lie 4\te !~ . If k ~ 2. , tt. ... G- ;, s..LJ.& t:J ?·D.
s"-f,~ tu c;.. AM ~ s,/ - .r~ Hc. ~ ~ ..J.. '=I, 1, . .. :lJ:. 'I 1<-::::. .

. .( -e....l.. H;. . ~ .s~- 6~ 'f· e a..J .s. G- .:. .s..t.J(... ~ '1-'1 · U ":;.- 3.
-~~/PI 13'1- 1· 6 lf,;J) 'f- '--:I= I- > H~" Hi ;,.. ~ s~.' -s~ "J H,: . S~ we. "-1
?!!/ I ~""'" -flA..J Hz_ w .strUt.. tJ ~~~ #A k. /1:- "chJ ~ ttJ G-

"'- ~ ·

~

L~

1·1 ~.:... ·

,.
I

Citation

Hall, Philip, 1904-1982, “Philip Hall Lecture Notes on Group Theory – Part IX ,” WUSTL Digital Gateway Image Collections & Exhibitions, accessed April 20, 2024, http://omeka.wustl.edu/omeka/items/show/10791.

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