Lecture4 Directand semidirectproductsofabeliangroups

Definition Let I be an indexand let Gi for it I be agroup withoperation iDefinethedirectproduct ITE of Gi ie denotedby II Gi or G x xGnwhen

is G ELGi withoperationGi ie Chili GiThi ie I

Theidentityelement is ei ice inverseofGi ice is gillie IForeachje I there's a naturalembedding G G IEGi

injectivehomomorphismgj to i 1 gj l 1

realizingGj as a nominalsubgroupof ItGi j placeTIG Gj ItgGi

There'salso a naturalprojection surjectivehomomorphism T G GGi ie g

herTj I Itisj G iWhenGi's are all isomorphic I 1 r wewriteGrforItGi

Definition AgroupG isfinitelygenerated ifthereis afinitesubsetAof G sit G A

theoremFundamental TheoremoffinitelygeneratedabeliangroupsLet G be afinitelygeneratedabeliangoop divides

Then G Z x74.2x x Inst for integers r so zen la l InsMoreover suchOr hi ins are unique

This r iscalledtherankof rProof Abeliangroups Z

modules

Sofollowsfromclassificationofmodulesof aPID

Lemme If mine IN z satisfiesgod min 1 then 4mn2 742 742Proof Consider 4mn21 742 742 homomorphism

a a modm amudn

Kerp amodm n I É In lolg is injective

But 74mnz 4m2 742 Soy is an isomorphism

Ca Everyfinitelygeneratedabeliangroupisoftheform

G Z ftp.rzx Alprisz xftp.zx tp z x

r epi p pi are uniqueuptopermutation

Example 74302 7 1002 74602 745oz

Pf 2 302 7 1002 7422 7432 7452 7 42 7257 Theyare isomorphic7 602 74502 7432 7 42 7452 7422 74252

Example Listall abeliangroupsoforder 8 9 727 22 7422 7442

1482

2 32 7432 4327 227 92

Recognizingdirectproducts

Def For x.geG define x y x y xy thecommutatorofxandy 1883Note if xy yx x y 1 g x y g gig gig

Gder G Exy x.geG iscalledthecommutatorsubgroupof61184,774or thederivedsubgroupofG 1843174

Note NottruethateveryelementofG is a commutatoritselfThis G is anormalsubgroupofG

g pofandGIG is abelian ble xG yG YG x G

x y xyG G yes

Lemmy If A is an abeliangroupand g G A a homomorphism

then G'skerf bac g x y xy 1

Thus 9 factors as G 6 6 A

Theorem Suppose G is agroupwithsubgroupsHandK suchthatis HandK arenormal in G anda H n k I

Then HK E H x K

Proof H K G Hk is anormalsubgroupofGConsider g H x k Hk

Ch k hk

g is a homomorphism 94h k haka 79 hi k g hak11

gChinakikahih.k.kz hikihakhah kikkhakihi 1But kaffithi eh kikki e kSo khakihi eH K lil is

g isclearlysurjectiveo kerf Lk h Ikki kek hetil

h h e knH i Sober9 1

9y is an isomorphism

Automorphismof agroupDefinitionFor a groupG denote Ant G 9 G G isomorphism calledtheanyopho

groupsuchisomorphisms are called automorphisms E1778 17 84ClearlyAut G is a subgroupofSG thepermutationgroupof G

Example Foreach gEG define 9g G G

Sg x gigThisYg is an isomorphism calledtheconjugationbyg

So weget amap Ad G Aut G calledtheadjointmatG to Gg Effort I

Claim Ad is a homomorphism KerAd Z Gcette

rofthegroupGgEG txe G xg gx

Proof Chet Sgh 9g09hIndeed Sgh x gh x gh

9go9h x Gg hxh'IghxtigKer Ad ge G Egged

V xEG Gg x id x

gx ex gx xg D

Def Im Ad S Aut GInn G calledthegroupof innerautomorphisms infactanormalsubg

By1st isom thm Inn G 6 2 G

Example Hut Mna y Hz Hz isomorphism

1

Ina a mod n godCain 1

Aut Mpa G4vFp

Semidirectproduct

Considerthesituation a H G K E Gb H K I

ThenHK hk heH kek is asubgroupof G

If K was alsonormal HK EH x K and H K commutehikehaka hikyyik.kz Sothe K coordinate ismultiplicative

Reversingthisconstruction

Deft Let H andK begroupsand g K Aut H a homomorphism

Definethesemidetprodt H X K HogK 14141 tobeH K Ch k theH KEKhi ki ha k h geek ha kik

Check hi ki chakaChsks hi k Khak ChsKsx

hi gki ha Kika hs ks Chi ki h 9kochs kaksh g ki heg Kikachs kkks h gki hagka hi kikksChig ki he gth sloths k kks h9th haFck g kachi kikksh k gli's h k h gtk elk's hi Kk i D

Thesets hit theH E HXk ish theK EH x k are subgroups

H is anormalsubgroupT H X K K is a homomorphism H Ker a

Chik in kTwowaysto rememberthenotation Q HX K H is a normalsubgroup

goopGroupaction G G X H X K H isanted onbyK

Proposition Let H e k begroupsandlet g K Aut H be a homomorphism

TFAE TheFollowingAreEquivalent1 Theidentitymap

betweenH x K and H x k is agoophomomorphism hencean isom

a g isthetrivialhomomorphismfrom KtoAnt H3 KQ H X K

Typicalexample

Ant Mna MnaMna x Nva

Eg E Ha Aut HaNva X Il Dan

Eg Orderpggroup p q 1Known 7 92 is a cyclicgroupoforderg 1

So it contains auniquesubgroupoforderp2 92 x Epa a groupoforderpg

Fact Fortwodifferentnontrivialhomomorphisms91,92742 044927

Thesemidirectproducts 7 92 xp Mpa are isomorphic

Allgroupsoforderpg areeither isom to7pqz or Iga x Ipa

Eg 7472 X7432 comesfrom 7432 HzO I 2 1 1 2 4 or 1 4 2

a b az.ba faitaz2b bitbor a taz 4h bitb