Notation and preliminaries -- Types and rankโ{128}{148}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