Free groupLet \(A\) be an abelian group. Let \((e_i)_{i \in I}\) be a family of elements of \(A\). We say that this family is a basis for \(A\) if the family is not empty, and if every element of \(A\) has a unique expression as a linear expression \[x = \sum_{i \in I} x_i e_i\] where \(x_i \in \mat……