Chapter 6 FINITELY GENERATED ABELIAN GROUPS

Question 1

A group is FG if

Answer 1

there exists S a subset of G where S is finite and < S > =G

Question 2

Zn x Zm is isomorphic to Znm <=>

Answer 2

hcf(n,m)=1

Question 3

Classification theorem for finitely generated abelian groups

Answer 3

Any FG abelian group G is isomorphic to a direct product of cyclic groups.

Zm1 x Zm2 x…..x Zmk x Z^S

where m1 | m2,….. mk-1 | mk and s>0
m1,…mk are the torsion coefficients of G
S is the rank of G

Question 4

Any finite abelian group G is isomorphic to

Answer 4

Zm1 x Zm2 x …x Zmk where mi | mi+1
i = 1,…,k-1
|G|= m1…mk

Question 5

Any FG abelian group with no elements of finite order (apart from 1) is isomorphic to

Answer 5

Z^S = Zx….xZ

S belongs to NU{0}

Question 6

Let
G1= Zm1 x …. x Zmk x Z^s
G1= Zn1 x …. x Znk x Z^t

where mi | mi+1, ni | ni+1 then….

Answer 6

Then G1 isomorphic to G2 <=> k=l, s=t ,ni=mi for all i