In this chapter we describe an application of Martin's Axiom in the theory of infinite abelian groups.

Throughout this chapter, a group means an infinite abelian group (with operation +). A group is free if it has a basis, i.e. a set of generators that are linearly independent.

Free groups are characterized by the following property: let A be a free group, and let π:B → A be a homomorphism of some group B onto A. Let X be a basis of A, and for each x∈X, pick φ(x)∈B so that π(φ(x)) = x. Since X is a basis, φ can be extended to an (injective) homomorphism φ of A into B such that π(φ(a)) = a for all a∈A.

Now let A be an arbitrary group.

Definition

A is a W-group if for any homomorphism π:B → A onto A with kernel Z, there exists a homomorphism φ:A → B such that π(φ(a)) = a for all a∈A.

By the remark preceding Definition 5.1, every free group is a W-group.

Whitehead's problem

Is every W-group free?

It had been known that every countable W-group is free. Shelah proved that Whitehead's problem is undecidable in set theory. Here we are concerned with the half of his solution that shows that it is consistent that an uncountable W-group exists that is not free:

Theorem (Shelah)

MAℵ1 implies that there is a W-group of cardinality ℵ1 that is not free.

We shall outline the idea of Shelah's proof, but first we need some more definitions and facts on W-groups.