Infinite Abelian Groups and Homological Methods

Abstract

This paper has two sections concerned with the characterization of the algebraic structures of certain types of infinite abelian groups such as compact groups, or tensor products of torsion groups. It also has a section which gives an approach to the problems of mixed abelian groups (those which are neither torsion nor torsion-free) by establishing a duality between torsion groups and those abelian groups which are as mixed as (that is, they are as far away as possible from being just a direct sum of a torsion group and a torsion-free group). There is a final section containing some miscellaneous results on torsion-free abelian groups. For the reader unfamiliar with homological algebra, we give in Section 1 a brief outline of some results and methods of that theory. Almost all our proofs will depend heavily on homological methods. Section 2 is devoted to the duality of which we have spoken. A reduced group is called co-torsion if it is always a direct summand whenever it appears as a subgroup with a torsion-free quotient group. In the same way that Pontrjagin duality reduces the problems of compact groups to those of discrete groups, the problems of co-torsion groups are reduced to those of torsion groups by a one-to-one duality between the groups of these two classes. A method is then given by which all the mixed groups with a given torsion group can be constructed from the co-torsion group dual to that torsion group. The main result of Section 3 is that an abelian group can be a compact topological group if and only if it is isomorphic to a direct product (unrestricted direct sum) of copies of finite cyclic groups, p-adic integers, the reals, and the groups Z(p-) where, for each prime p, the number of copies of Z(p-) does not exceed the number of copies of the reals. It is also proved that if G is an abelian group with the multiples n! G considered as neighborhoods of the identity, then essentially G is a direct summand of a direct product of finite cyclic groups if and only if it is complete in the metric defined by these neighborhoods, while G is a direct sum of cyclic groups if and only if it is as far from complete as possible in this metric. At the very end of Section 3 it is shown that a torsion-free group is a direct summand of a direct product of finite 366