How to prove an Abelian Group is not free?

[gunbuddy]

Hi,

 

How can I prove a certain abelian group is not free?

If the given group is small, I can consider all the possible subsets of the group which are candidates to be the basis of the group and prove each of those cannot be the basis.

As an example, if I consider the abelian group Z₅ with addition, should I consider all the subsets of Z₅ and prove that any of the subsets cannot be the basis?

[jsaacmk]

You need to work with the precise definition of your group to determine whether the abelian group is free or not. Your approach is appropriate since if there is no subset of Z₅ that is a basis, then Z₅ is not free. Do you care to share the abelian group you are working with?