Finitely Generated Abelian Groups

G t Abelian groupAdditive Notation i e G ne ZGiven

x x

ux µt Cn times if u o

c e it n o

y K t C K C u times 17 Uco

CareFall inverse atusually written x H

not multiplication

H C G subgroup REG at H a th the G3Always normal

at H gt H x y E H

Aim Classify all Finitely generated Abelian groups up towe say x is a

isomorphrsin torsion element of GE

Definition t G i a E C I order c 3C G

Proposition EG C G is a subgroup called the Ension subgroup

P

x E t G F u e IN such that ax O

ord o L o e tax ye TG F m u C IN such that ur a O try

Abelianmu sexy n mx t ul uy 0 0 0

set ye toK E CG 3 me IN such that ma o mtx o

seeta IIExampleG Rly 1 x E th F me IN such that

u a Cnn Co Fire IN such that n c c 7C x E Q

EG Iz 2

Remarks

If G non Abelian EG may be a subgroup

Definition

G c's torsion G tG

G is torsion free e a 903

Example

Ifl co G torsion

z t torsion

zu c torsion tree

Proposition G 4G torsion Free

Proof

Let atte be torsion in Etta3 he IN such that nx tE Ot Ct

ux EEG3 in C IN such that ur use O urn x o

x EEG x ta o taD

f gDefinition

A finitely generated Abelian group G is tree it

3 se au E C such that the Following property holds

Given get 3 Xie 2 Such that

g T d t 7 K t t Xu Ln

we call x du a Z basis for G

Recreate

Ex seu C G a Z basis ord sci a fi

and G gp Ge gp Gcn ten 1

po Let G be a Finitely generated Free Abeliangrop

Any two Z bases have the same size

Proofbe Z bases for G

et Ea du and Ey you

Define ZG i 2g I g e gnormal subgroup of G

ZG X x 1 Xu Xu I 217Given a b e G a di di t Lu Xu b B K t Baku

G

ZG b ZG a b Li B x t t Chu Bu an C 2

21 Li Bi Hi

1 4 2

Exactly the same logic applied to Ey you implies

Eta z'm 2 _I n ma

etinition timing generated tree Abelian

ankle size at any 2 basis Analogue atdeniension cisTheorem Cui ear algebra

G Finitely generated G finitelygenerated andFree Abelian torsion Tree

Loot See Notes

em G finitely generated Abelian

Gf Finitely generated Free Abelian

ProstgPKK xD G gp Kittu xu tG Etta

Hence G Finitely generatedG tu Finitely generated

G AG torsion Free Alta tree Abelian a

Definition Finitely generated Abelian

rank G i Van KC G ta

then Let G be F g Abelian with

rank G u Then F F C G a Finitely

generated Free Abelian Subgroup such that

G F TG and rankCH a

Zoot

Finitely generated Free Abeian I an C G

such that x EG Ku th is a Z basis For Gtaet F gp Gx an

Clair a Xu CF is a Z basis for F Ex Cta sanctaa Z basis For

X x In i an t 1 Tu an G LEG

X Cx 1T G t 1 An x te 7 x ta t X Kutta

Xi Xi f i v n is a Z basis for F

We must now prove G F to Note G Abelian 2 holds

We should check 4

Let gt G DX Xu C 2 such that

g TG X la G t An x te

Ex c Xuan G

g x x t And u t h where he t G

E F C

tatth Iz the 7 Fz h Ln n n n Fis n aF ta F th t.rsjuutret F ta

oud 7 7 C a 71 72 0 71 72he he

a F EG I

Cuy Let G be a Finitey generated Abelian

group then G Ex EG where

u rank G and It G Is so

G F EG hence we can define

the homomorphismG EG

7 h bein aF EG

Imf EG kerf F GIF TG1st IsoTheorem gp

x xn ta

tG Xie 1 AuduG 7 g GIF t g EG 7g yG T g and torsion leg a

o i Eordki

ankle u F Z

G F EG Z EG

f gFreeTAbelian finite Abelian

Hence to classify F g Abelian groups we must

now classify all Finite Abelian groups