Lecture 2 Corsetsandquotientgroups

Parallel developmentofmathstoriesVectorspares Groupssubspaces subgroups

directsums directproducts v

linearmapslinear isomorphisms

homomorphisms

isomorphisms

quotientspares I I intgroupsf V W V kerf W Isomorphismtheorems

V Velu Velu nVa

Definition Let H be a subgroupof G Aleftcoset 7 13 is a setoftheformforsome gEG gtl gh heH

inparticular ifge ti gH H

A rightcoset is a setoftheformforsomegeGHg hg heh

PropositionTwo left corsetsg Handg H areeitherequal gig EHor disjoint gig H

Proof Needtoprove GH gaH 0 gig EH g H g H

If gigEH g H g gig g H

bleanyelementheH hH H

any

If gang Htt sat tI my sheet DThen g xhi g xhi gig this xhi

Definition Write G H SH I ge G forthesetofleftcosetsSimilarly HG Hg gEG

isgHGg qgH

is a disjointunionThe

cosetcontainggeG

gegH if g getCall 6 4 G H theindexofHas asubgroupofG

Lagrange'sTheorem If G is a finitegroupand H G then H G

Proof In fact GH GAH D

Cox as If G is afinitegroup then 1 1 G

Proof 1 1 x G D

z xG

e

Example G Ha amodN la N 11 6 4 NThenforevery at Ina all 1mudN Euler'stheorem

Coz If G p is a prime then G is cyclic abelian

ProofTake e G thentxt G p in p G x

50Definition Let G be agroup a.geG gag iscalledtheconjugateof a bygLemmeIf H is asubgroupofG andgE G thengHg is a subgroup

ghg't h calledtheconjugateoftibygProof Given gag gbge gHgt

gag gbj's gag 5 b g gabg e gHg V D

DefinitionAsubgroupH e G isnormalif tge G H gHgie allconjugatesofH arejustHitselfEg H Hg soleftcoset isthesame as rightcoset

WewriteHQ G toindicatenormalsubgps E.g I OG G OG

Inthiscase for a b eG alt bH I ke theatt le bHabHH abH

ie we have a welldefinedgroupstructure on G HTheidentity is eH H theinverseof alt is a H

Wecall G Hthequotientgroup fatorgrop FatG G H is a surjectivehomomorphism

Y a aH

Sometechnicalconstructions

Proposition Let H Kbesubgroupsof a groupG define HK hk heH ke kWhenG isfinite Hk Hh

Hnk

Proof HK is a disjoint unionofleftcosetsof KHK hi k whak w Whnk

Claim Forthesamehi H h Hnk u Whn Hnk

Thenn th

Hnk

Proof Foreveryh heHhk like Whe k shik eHn ke h Hnk h Hnk

So Hk heyhk h k w wh K

H Gh Hnk h Hnk w whnHnk D

hellRmt Hkneednot be agroupabove

E.g In 6 53 H iz K i3 then Hk H k 4Hkcan'tbe a subspofG ble446

Lemme If HK KH as a set then Hk is a subgroupof GE ie everyproducthk canbewrittenas kik andvice versa

e.gWhenH is a normalsubgroups theH hk Kh so HK KH

Proof For hi h EH ki kaE khik haka hi

kikffpI h.lk eHKD

Lemma If bothHand K are normalsubgroupsofG thenso is HKProof AgeG gHk HgK Hkg D

Definition For a homomorphismY G Hofgroupsthekernel is Kerg ge G gig en

BEASTLea asTheimage9 G is a subgroupof H

as Thekernel kery is a normalsubgroupofGProof i Note 9 g 995 g g g gi g gg e g G

a If g g eKerg then 9 9,95 g gi g gif en ef eng gi eKerg So Kerg is a subgroup

Foranyg eG gekerg g ggg gig'sg g 91g'sg g englg j en

So ggg eKery Keryisnormal

Lemma A homomorphism 9 G H ofgroups is injectiveifandonlyif kerf feel

oy 9 ofgroup f y 931Proof injective keng leg isclear as glee env

Conversely supposekerf eat Wes y isinjectiveSay 91917 9192 forsomeg g EGThen 9 9,95 919 ggzJ en

gig ekerg leg g gie gig D

TheFirst IsomorphismTheorem

oy

If 9 G H is a homomorphismofgroups thenkerf IG and Our4 916Picture ProofDefine amap 4Oberg 0916

gLEMG gkey to 9 gas4iswell definedI dif g Kerp gkerf 919 ckerfLEIGH then 919 g 92.919

g g 91919 Ig gaa Y is surjective b c every

elementofg G takestheformoffigit'stheimageof gKerg

3 4 is injective Enoughtocheck kerf hergThis isbio if 4 gKerg glg e gekery

gkerf kerf So kent kerf4 4 is ahomomorphism 4 gkerg gakerf 4 9 gherg 919,92

4 Sikery 419key 919,19194 D

The SecondIsomorphismTheorem SlightlyweakerthantheversionfromthebookLet G be agroup and let AE G Be Gbesubgoap.seonly

need AnormalizesBi e HaeA aBatB

Then AB is a subgroupofG B QAB A BIA and

ABIB E AA BProof Haveprovedthat ABE GFirstshow B Q AB given aEA beB abB ab abBb at aBat BSothequotientgroupABIBmakessenseDefine a homomorphism g A AB ABIB

a to a to aB

g is clearlysurjective bic abB aB g a

I s gKerg a e A aB B AnB inparticular it'snormal in A

ATB

Bythefirstisomthm A A B E ABIB

Rmt GAB

A

yhavethe samesize

1GTheThird IsomorphismTheorem1 ELet Gbe agroupand HandK be

normalsubgpsof G with H EK K GHGThen K HIGH and GH K H Gk H

If wedenotequotientbyH by a bar thissaysGK GKProof Consider 4 GH GK

gH ing k

G iswelldefined if g ti g H then g g h9 K g hk g K

Y is a homomorphism g g H 9 H g g g H 9,9 K

g gH g H g K g k 9 g k

Y is surjective clear

kerg SH stgk k gh gek KA

kerg SH s l gk k gH gekf 1 4Ige k

So k it is a normalsubgroupofGHThefirstisomorphismthm GH

KH 4K