Publisher Summary The mixed groups A form the most general class of abelian groups and a satisfactory structure theory must provide full information about their torsion parts T and the corresponding torsion-free group A/T, and at the same time describe the way in which these groups are put together to form A. Therefore, such a theory can be expected only for those classes of mixed groups A for which both T and A/T can be characterized in a satisfactory way. Then, the problem reduces to finding tools by means of which the nonisomorphic extensions of T by A/T can be described. The height-matrices discussed in the chapter are adequate for several groups of torsion-free rank I. A great deal of work has been done on the splitting problem—that is, when the mixed group splits into the direct sum of its torsion part and a torsion-free group. The chapter describes all torsion groups T and all torsion-free groups G, respectively, such that every mixed group splits whose torsion part is isomorphic to T and whose quotient mod its torsion part is isomorphic to G, respectively. The chapter also explains the construction of groups with given Ulm sequences. A rather intricate procedure leads to an existence theorem that is useful in establishing the existence of groups with certain properties.