Philip Hall Lecture Notes on Group Theory – Part XIVb

Philip Hall Lecture Notes on Group Theory – Part XIVb

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This material is a set of unpublished, hand-written lecture notes composed by Philip Hall (1904-1982).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


§14 Some Special p-Groups (pp. 123-130)

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.

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lr I G-: J-t;./ , lk '"~ 1' ~~ ~~ 1- G --~
~~ G-/G! L·~·~ • He...u.. ~ ~u..tr.

u..

.L,
l.1k

~
JJ...

~

If

C

=H

Theorem 14.3
Suppooe

implies that
C

C =G.

that

, it is an easy consequence of Theorem 14.3 that

::>

H and

Let

G1 =

p

is odd •

~(G).H •

c ::>H

)

S
O8 Qs'
8

G

=

s

s' z. , where
1
If

= ~ (C).c(.Q 1 (C))

subgroup of

C( 121 (c))
tY

~

, P

~

Since 4t/C

,

Q·

~

•

Then

•

is cyclic, Theorem 14.3 implies

By Lemma 14.41,

C

=2

l Hl /

2 •

z1 = .C22 (H)

08 Q
8

and . H •

0

p

am

from Theorem 14.1 , so suppose
1 (C

is one of

=2

G

= G1C(G1 )

and so

G

= G1

•

G = Pr H , and we are done.

By Lemma 14.4, we have

Finally, suppose

G

p

=G

If

IH
l

Since

•

== 2 ' the :i::elmll

C1

Thus,

Furt~ermore,

It follows that

theorem follows

is of order 2 , it follows that

C is the central product of

, the proof is complete.

by Lemma 14.41.

•

•

C( ~ (C))

Otherwise, we get
H is a maximal aorma.l abelian
is one of

The various ismorphisms follow easily from Theorem 14.4.
A

P k, Q k, 0 k •
2
2
2 .

.

.

r

~~ "- ;- ~ · ~ )H-~. 1' ~r
._J u z /,c_ ~clu 1= {!}.
~A tt.. ~,.L J>~c.t Hz ~ ~ &- -:11~
~~"v ~ hc.c.-o.e- ft.. i.tL-/~ . .1~ 'c - ~ k Z, = '1}1-/ wt1..
z, = {J'I'~-'1 a.J 1t.. o-.4-~z.u- 1-.. z,· · ~ ~.2 w A..t-z,.

LG- J Let H
~ ; .., z

~

L..d

14-·S .

G- k..

"~-MeL.·,."' ~- ~

o..

~ ~,..,..~t_;, M~ .r~ eJ. G- L.

s~ /l..d,

" · ke.. ~J ld- Z ~ {'!1 -~ G k
e

7 o-rJ.c.,. ; ~. 7L.JI4.. .
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\fl)

If

;

~.,.. ~ 0 ~.,._,
Ck~

1\o

= 2.

4"J

(..,..>- o)

o-r

~ &/ /l.....e,

0;z4:

IC'\r

"\

~

If\

w~•

41'<.

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'ilc 0 (Q,

,(\

0.z.lc 0

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(lii)
4t..J

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~

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o-cJ "'~ I ,

it...

a-

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02.11\Y"-2.-

~

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.,_ '-'C.tlc
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.

02.1C 0 3 4.
.
A'l)*-J

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t... ...

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•

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£Je. ~ ~1:
l~l~t-·s'
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~ G.' ~ ~eli.e. .

I

c.c.4-~ ~ of ~ 1 lie.
~· ~,.. _, ~.z." ~..,. (A >3 ~ r-~ o)

o.or-l "" ~ 0

~.. 6('t'" ~
'-".z.~ ~

u. ..

G-

h\-:a I "

,...,

• - .

•

z ~.,. ;w. J""-&.

.,.. 7 0

Z6(-~ zo~.,._, ':;t zo~~'Y

1 -s~ s.....u..

·-·

a.. ~J..:,.

tt..J: l G-'

P~ ': Ld z-= J- t;.'. s~CA. .z ~ G., z ~ c;.'. ~ ~--" ~~
J\Ot• ..1,
~d.:.c ./~rhf H 1 G- ~c.L ;. ~...:..t ·..sw.J.j._cC t-

-:s: H ~ G- 1 • If G-' . M ,.._.t ~ck.c. /t:c., H.< er' ~ L, s:~ li >,.
/1.».. ~ a.. ~~ .t~~ I< -r G. s-.J. /J,.J,; H < K. s G-. 1 -~J
IK ~ HI-=-;. n.... K Co )\,f; CAJ~ • II I< it, MeL.~ it .:- 'I iaJ'L
("', 1) ~ /HI-=-;,... , ~~ .11..,1< ~ -1' ,.,..J,u. p2.. . If I< ~ 1\o-&
A-£.eL.4... :> Jl 1 1< iA~filL 'f ~r /'~ ~ ~ ; ~ l:J l'f · 3 (l) ) a-.J -J..D - r
'I
~ /> = 2. ;~kJ ~ (;, 1 ~ 2., ~J lcr·.J (!v) • 7k. ~ ~~ ~
u frr /<. lr lee.. ~ "f /L_ ~~ 0l.,...., ~ ~ ...,,., , ~.z....'*' /,'J l't-·3 <til ) o.J(v) ~ ~:

Z

:1

j

w-~ · 4. .t~ ·L
L ~ }z. t;. a-J &o [L, G--'j -: 1.

B-t I<

1. ·

.

...J...-1~1&4J ~~G . . ~
l(f "f· ~ C!_i"J. ~u.. L ~ }K • B....t ~

7

13o

~

q _,.., , .. . 4ve.

C4v't~ lk.k I<

a-J

~

~-c. "/- ~ .l , s. K.

/r..J: } 6-'

'1- tl.uil- .

, [M , G-1] = 1. . 7L:.

l.h..

7k... /i<JG-

e,.J-~-J;, 11;,_ t~U.q_,t~

<:! d.: c... •

&.

f~

~~ ~

~ c.~t:.~ J~ M=.n.,K 'I~ }"L·

~ w-e ~ ~t ~~"

to ,

C..rtllt.-1-

.S..;.CA.. J-G- 1 i., A. ~!..--;.. /tleL;... s~ '1 G-,
c.t ih "'J~ I 1-3 ~~~ • . f.k..CL (;./ ;. 4./ckc.. LJ l'f-·SI' a-J ZG' ~ A..
e.l....xa ~ 14.:.. ~ ~ ' H. G- • s. z c;/ ;. -r:-- G.o • 1..«.1:- H A<- A.
c.. t.rocw
'· · -'··c.
c.Jv.-..~~
.,

1

lit-·!>.

~...:..L L'3c.L:c. .(~ .-1 G- eq.,J;,.,-.~ z G-'. ~~ ~ H < G,
a-.vi if C -:. CG- (H), /A,. ;, C ;,. " ~~ Al,<l.·a.. .s~r-tltAf 1 Gc,.,J-~ fl ·~ k-c.c.. J- C ~ 'teL:c. , ~J .r. J-C. -=H .

If

1k C ~ 1 cl.a.u

C ,_ H,

t'J 1·1 Ul _,(iiJ.
lj

~ -~'.a.

:?;:

•

.~

~.

[ f,' .) 'fc: J

,

....~

2

At,., C/ ~ G-/ ~ H ;::.A-C.

~sJ ... [f,r,J.,..r -for «11. {,' ~

ftr,

l..d . 14 Ak£;_ frt'? C/H k 1 'i~

w '-t

H! I

= [ r, > fc:r~~1 '::.

,

Hl ~

.) .-- k

~ .: =

1

I
M\H.

c.;H .

l Mk 1

a.

2. ,3.) --

/> r,- •

a...J
,.,., -•

b

-1UACL

t;

l

r.t, · --) ,

7l..t,~

1,

,~-2..

E

w<

c,

~

~

H -~c.

.

t.,c J =- f'fc: , t: J = :l frr
All.. i .~j-.
If L/H = v;,:..., (C/H), d {:ft ~vwlt;..t L cLr ~ , L/-== L , '- /H:.
~ L 7chc.. , ~~":/ k ft.. c.Lh.u.. "' H.. '*-'u. r 1 = .l ~J C./ H .:..
~~' cw..J .to C 1 &- 1- ~ !·
J.k..~

fj

-=

r:a. •

/f

1j

~

I ,

we.

[

fi

>