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Unformatted text preview: 195 3.6. FINITELY GENERATED ABELIAN GROUPS P i ri xi . Let N denote the kernel of ' . According to Theorem 3.5.13,
N is free of rank s Ä n, and there exists a basis fv1 ; : : : ; vn g of Zn and
positive integers d1 ; : : : ; ds such that
fd1 v1 ; : : : ; ds vs g is a basis of N and
di divides dj for i Ä j .
Therefore G Š Zn =N D .Zv1 ˚ ˚ Zvn /=.Zd1 v1 ˚ ˚ Z ds vs / The following lemma applies to this situation:
Lemma 3.6.1. Let A1 ; : : : ; An be abelian groups and Bi Â Ai subgroups.
.A1 Bn / Š A1 =B1 An /=.B1 An =Bn : Proof. Consider the homomorphism of A1
An onto A1 =B1
An =Bn deﬁned by .a1 ; : : : ; an / 7! .a1 C B1 ;
; an C Bn /. The kernel of
this map is B1
Bn Â A1
An , so by the isomorphism theorem
.A1 Bn / Š A1 =B1 An /=.B1 An =Bn :
I Observe that Zvi =Zdi vi Š Z=di Z D Zdi , since
r 7! rvi C Zdi vi
is a surjective Z–module homomorphism with kernel di Z. Applying Lemma
3.6.1 and this observation to the situation described above gives
G Š .Zv1 ˚ ˚ Zvn /=.Zd1 v1 ˚ ˚ Zds vs /
Š .Zv1 =Zd1 v1 /
Š Z=di Z
D Zdi .Zvs =Zds vs /
Z=ds Z Zds Z ns Z Zvs C1 Z vn ns : If some di were equal to 1, then Z=di Z would be the trivial group, so
could be dropped from the direct product. But this would display G as
generated by fewer than n elements, contradicting the minimality of n.
We have proved the existence part of the following fundamental theorem:
Theorem 3.6.2. (Fundamental Theorem of Finitely Generated Abelian
Groups: Invariant Factor Form) Let G be a ﬁnitely generated abelian
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