The Implications of Linear Independence on Free Abelian Groups

Abstract

The existence of a basis in free abelian groups could not be separated from the concept of linear independence. In vector spaces, linear independence has several implications that play an important role in various basis discussions. The discussion regarding the free abelian groups focuses more on its basis and implications. In this article, to get deeper implications regarding the basis of free abelian groups, namely the implications of their linear independence first, such as if a subset X of F is linearly independent then every element of the subgroup generated by X can be written uniquely as a linear combination of some elements of X, and other implications. The methodology used in this article is literature study and focus group discussion to obtain information from algebra experts. The result of the research are there is an implication of linear independence in free abelian groups that applies as well as linear independence in vector space, but there are also certain implications that cannot be applied to free abelian groups. This will be interesting as further research regarding the properties of the basis in free abelian groups.