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How to prove an Abelian Group is not free?

gunbuddy
in Linear Algebra and Group Theory

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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?

  • 3 months later...

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?

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