Qs II ti j kLift i2 j2 k I C Die i

ij ji k

Q8T

c

notohm.ct.rs

csifgTck7all normalIc is c

chractiska

TodaylstopniGroupact

Ded let Gbeagraxp.X.atA left action f Gen X is a tuple G X a

aiGxX X

g s g s algssuchthat AgheG X gChx lgh x

ex X

Altrnakhay.ge X is a G Set

X X a

A homomorphism of G Setsis a map

fatsf X y sif g x g fld

isomorphisms automorphisms epimonphisms

style hey

mm mph.sn

ligatesub actors sub G sets

DefI let Gha gorp X set a G set

is a homomorphism G S q X XIq isabject

Gun a G S

A G sya km ofGSet s from

x yis fix 9 set

f algCx p g fix

Note categories of G letss G Set's

are isomorphic categories

GSet G X a

G set

g x algikG Sxg x g x

G 4

EximGatson ittf vi left

molt G X

g h gh G SGij Cayley'sthin

Trinal actin on a atxg xx G S

g e

CosetactsG acts on X HH gH l ge G

g g H gg H

G SGH

Conjugates eaten

G acts on G via g h ghje inng h

G SGr Caj actin on

subgroups

A SubCGI HAG

g H gHgt

Automorphism actin on G

Aut G acts on G

q g qCgl

Aot action on subps

At GP Sob Ginacts on

y H put

Specific actors

Snacts on El a

Dan acts on El n votes on n g on

Dg B also acts on 2 dgauls

2 21 Ca ads on El a

Ghul d acts on 6

121 If G acts on X Xdeke stabbed go.co gx x

stabola G

BI If X is a G set defe landoftheactor kerCD kr G Sx

Xo G

121 If Gatson X X we date

theorbitof x written Gx gxlgc.ES

Note Actors gie eq.netuus

X y of y gx songc G

orbits eq classes

alsoget grapoids 7

gag

Prepe Suppose GPX and greythen Staboly gstabolagPI 2 dearsestabgcxkblgsgY.y lgsgY.gx.gs gx y

t

IactionCgsgtg x

E gx y g y x

pun g stably15T'sstake

skhgcpe.gs4dxlgD

Dd Tranday goGI guy

Prep If y gx thenTranday gstabaki

PI 7 deare nate if tetransday g testify

C c

gstabglx.D.cmTransolxsyY 1staboCxyif y cGx

theorem Orbit Stabilizertheorem

aka most important theorem inmeth

suppose X is a G set a X then

16 1 Gi stab I IGI1stab l

PII F syeate m pG 93 Gx

g gx

Fr y c Gx y Cg TransdayI g Gil Istanbul

because molliplraton IGI I Stabokil16 1afthemeanzafthewnd D

G 4h left nullsylenbit

StabelH go.co gH H3iH

thin 14th f Lagrange

GO sub G conj

H G Stadt geG gHg'tHIN

NGCH normalizer

largestshypncoutacy Hart Hokk

Props Suppose He G s I Koto D IHI 7

then Hob

PI consider GOGH by leftmolt

Get G 4 Say and kerqcskbgH.tl

ly 1st iso Gharg L SGay

14kg111Saul G HI EGHI mme Girl c I

4 17 in with HH

14 Izu Kl m

1info are largeH

that'm

Hilary m

assumption IHIpinetum

Hilary I lance HOGD

Ded Cott gcGIga agallao.AZA CG subset gag sa

GStaboldairacoyaut

DI 2CG CglGKer inn G SG

2 G a G

Dd g Iga are conjugateif the Gsk

hg.li gaCi e sane orbit w conjecture

Cajugayclasses are magical

G U conj classes w e disjointunion

Warbikundrcanj U union

choose representative of conjclasses

Zi ar g sgmwhelcojcless.fze.tl

cojcluses.fgil I

2 G 7 Zr

Kaj cha f g I 141stabolgill EGCdgis

theorem Theclass equation lkeyto power

IG 12441 t.IEEG Cdgisg gmreps In y deses

neutral