Infinite abelian groups: Methods and results

Abstract

Publisher Summary This chapter describes the methods and results of infinite abelian groups. A group is Abelian if the group operation, usually called “addition,” is commutative. The theory of Abelian groups is an independent branch of algebra. Each finite Abelian group is a direct sum of cyclic groups of prime power orders. The theory of Abelian groups breaks down in three main parts: torsion groups; torsion-free groups; and mixed groups. In the theory of torsion-free groups, the notion of rank is essential. The rank r (G) of a torsion-free group G as cardinality of a maximal linearly independent system of elements of G. Torsion-free groups of rank one have the simplest structure theory. Each such group is isomorphic to some subgroup of the group Q. Some classes of non-splitting mixed groups are described by means of invariants. The Ulm theorem describes all countable p-groups by means of numerical invariants. Some other group classes are also described in terms of invariants. A torsion-free completely decomposable group can be quasi-isomorphic to a torsion free group which is not completely decomposable.