CHAPTER VII - TORSION FREE GROUPS

Abstract

This chapter discusses torsion free groups. The problem of determining all torsion free groups is more difficult than the corresponding problem in the case of torsion groups, and, up to now, only for comparatively restricted classes of torsion free groups does there exist a theory as satisfactory as that of Prüfer–Ulm–Zippin. A satisfactory complete system of invariants is known for groups of rank 1 and for their direct sums, but in all other cases, no honest invariants are known. However, certain schemes for construction of torsion free groups are known, usually in terms of matrices. Furthermore, the groups obtained by these schemes are isomorphic if and only if the schemes are equivalent. The chapter discusses a scheme for countable torsion free groups, while another scheme for torsion free groups of arbitrary power may be found as a special case of a more general theory. Torsion free groups that behave in a special manner from the point of view of direct decompositions, such as the completely decomposable groups and the separable groups, play a distinguished role in the theory.