Divisibility and Injectivity
Most perfect objects in the category of abelian groups are those groups in which we can also ‘divide:’ for every element a and for every positive integer n, the equation nx = a has a (not necessarily unique) solution for x in the group. These objects are the divisible groups which are universal in the sense that every group can be embedded as a subgroup in a suitable divisible group.
The divisible groups form one of the most important classes of abelian groups. In our presentation, we focus on their most prominent properties, many of them may serve as their characterization. Their outstanding feature is that they coincide with the injective groups, and as such they are direct summands in every group containing them as subgroups. Moreover, they constitute a class in which the groups admit a satisfactory characterization in terms of cardinal invariants.
The concluding topic for this chapter is concerned with a remarkable duality between maximum and minimum conditions on subgroups.