# The Grothendienck Group

Desvl at
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……