Post on 04-Oct-2021
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