We state and prove a sufficient condition for finitely generated Abelian Groups to be the direct product of its generators, and state a counterexample to the conclusion when the condition is not satisfied.
Let be an abelian group and .
Suppose the generators are linearly independent over , that is, whenever for some integers , we have .
(Here we are using additive notation for , where the identity of is written as 0, the inverse of is written as ).
Define the following map by
We can check that is a group homomorphism.
We have that is surjective since any element is by definition a combination of finitely many elements of the generating set and their inverses. Since is abelian, for some .
Also, is injective since if , then all the coefficients are zero (by the linear independence condition). Thus is trivial.
Hence is an isomorphism.
Note that without the linear independence condition, the conclusion may not be true. Consider which is abelian with order 30. Consider , .
We can see that , by observing that , , . However has order 90. Thus .