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