Conjugation action of G on itself.

Stabilizer subgroup of

XEGis Cacx ) = ( ge G :

gxg- '

=x ) centralizer subgroup of

G .

Orbit of X C

is Kacx ) = ( gxg' '

:

g EG ) is the conjugacy class ofx in G .

ZCG ) is kernel of action :

Note ZEZCG ) ⇐ gzg'

=z for all g EG

⇐ Conjugacy class of z is Ez)

Thedasseguatiori

Let G be a finite group and gig . . .

, gr EG

representatives of the distinct conjugacy classesof G not contained in

'

the centre ZCG ) .

Then lol = 17h01 t E. [ G : Cacgc ))

Proofs In any group action of G on X

we have X = G . Xi where Xc,

at,

. . .

,s

are the distinct representatives . of the G- orbits on

×Orbit Stabilizer Theorem says

.la.xl=[ G : Gx )

So lXl= I,

[ a : axe ]

We have X -

-G and Gxi = Cath

This would be sufficient if it weren't for

the IZCOII term.

But each ett of ZCG ) is In its own conjugacyclass ( orbit )

.

So suppose ZCG ) -- Lz

,. . .

Ztt

and the other representatives of thedistinct conjugacy classes are Cgi ,

. .

, gr )

thenG= If ,

⇐ I U Kacgc )

⇒ lol = E. I t E,

[ G : Cacgc) )

= IZCGH t €,

[ G :C aegis I

F-tya.me#willshowlKoCdl=k.i.nm!iI..ksl.msksif o has cycle type mi "

Z . - . 2ms"

including I cycles .

S4 has cycle types :

(4) : 6

C3,1 ) : 8( 22) : 3

( 2,17 : 6

( 14 ) : I

17h54)

1=124--1541--12-4,54

) ) t Gt3 t 8.16

theorem If p is a prime a P is a p -

groupie IPI = pa for Sore azl

then IZCPH > I

IPI = IZCP ) I t I,

LP :

Cpcgc) ]

where g. ,. . .

, gr are distinctconjugacy

class

representatives not in Zca )

lkpcge) I = [ P : Cpcgcj] 71 a [ P :

cpcgcs.tl/PllPjla--lZCP)/i I,

[ p : cpcg, ]⇒ p

I [ P : Cpcgc, ]

TI divisible by pBy equation IZCPH = O mod'p

So IZCPH= I is impossible .

So IZCP)17 I ⇒ ZCP ) t Ll ).

Lemme If GIG ) is cyclic ,then G must

be abelian.

Cyzca , =LgZ⑥ 7 for some GE G .

Let x, ye G ⇒ XZCG ) -

- g" ZCG )

YZCG) =geZCG)

So x = gkz y = glz '

Xy = gkzglz'

= gktlzz ' =gl' - ' '

z' z = glz '

gkz.

= yx .

Corollary If IPI =p' for some prime p

then P is abelian .

In fact P IZlyprz,

Or Z4pz×4pzProof . Since ZCP ) t I then IZLPII =p or p

-

If IZCPH=p then Hunt =p ⇒ Plzcp) I Cp

⇒ P is abelian .

So all groups of order p'

are abelian.

If P has an ett x of order p'

then 47 s P a thx > I =P ⇒ p=Lx7

Otherwise, any

I #EP must have order p .

Let y E P - Cx ?

< x7 E Lx ,y7EPKx, y ? I =p'

and so ex, y > =P

±

Cf : Cp x Cp → P Check this is

exa , yb) → xayb a groupiso

It is clearly a bijection

Fritsylowtheorem-i.fra prime divisor p of 1Gt

,a Sylow

p - subgroup is a subgroup P of order p"

where I Gl=pi 'r and Ptr .

That is p"

is maximal power ofp divisor

of IGI.

)

Firstsylowtheoremforevery prime p I IGI

,G has a

Sylow p- subgroup .

Pro Proof by strong induction on nzl.

Pln ) : If G is a groupof order

pnthen G has a Sylow p - subgroupPll) -

If IGI =p then GE Cp and

so a is itself a Sylow p- subgroup .

Assume Pcm ) for all mlnLet IGI = pn and n=pa

- 'r ptr

Corset pl IZCGHThen 7h01 has an element of order p

by Cauchy 's theoremN = Lx ) EZCG )

Then NEC

1%1 = pair

So GN has Sylow p-

sabyp Pym

where NEPSG⇒ IPINI =p

" 'r ⇒ 1Pl= par

Case I px IZCG )

Kl= IZCG ) I t EI,

[ G : Caged CH

for godistinct

representativesSuppose p

I [ G : Cacgc) ] for all i of conjugacyClasses .

not

and pl IGI in ZCG )

Then p I IZCGIISo by contradiction

, pH [ a : Cacgc) ] for some

But then lol = lcacge ) I Co : Cages ]t -

Recall lol = par rptr'

Then if m = [ a :C Cgc ) )gcdcpgm ) =L C since gcdcpgm) =p

's

Absa ).

a pxmBut then in

pal lcacgc ) I [ G : Cacgc) ]

⇒ pal lcacgcslSo I Cacgc) I = pas

lol = lcacgo ) t [ C : Cacgc) ]par= pas rn ⇒ r -

- Sm

& ptr ⇒ ptsSo lcocgc )l = pas pts

kacgcsl a lol⇒ By inductive hypothesis , Cacgc )

has Sylow p- subgp P of size p

"

PE Cacgc ) E GP is

.

ii. so Sylow p - subgroup of G .