Notation and preliminaries.- Types and rank¿1 groups.- Examples of indecomposable groups and direct sums.- Endomorphism rings and decompositions of rank 2 groups.- Pure subgroups of completely decomposable groups.- Homogeneous completely decomposable groups and generalizations.- Completely decomposable groups and generalizations.- Additive categories, quasi- isomorphism and near-isomorphism.- Stable range, substitution, cancellation, and exchange properties.- Subrings of finite dimensional Q-algebras.- Orders in finite dimensional simple Q-algebras.- Maximal orders in finite dimensional simple Q-algebras.- Near isomorphism and genus class.- Grothendieck groups.- Additive groups of subrings of finite dimensional Q-algebras.- Q-simple and p-simple groups.