Title:

Free groups and the axiom of choice

The NielsenSchreier theorem states that subgroups of free groups are free. As all of its proofs use the Axiom of Choice, it is natural to ask whether the theorem is equivalent to the Axiom of Choice. Other questions arise in this context, such as whether the same is true for free abelian groups, and whether free groups have a notion of dimension in the absence of Choice. In chapters 1 and 2 we define basic concepts and introduce FraenkelMostowski models. In chapter 3 the notion of dimension in free groups is investigated. We prove, without using the full Axiom of Choice, that all bases of a free group have the same cardinality. In contrast, a closely related statement is shown to be equivalent to the Axiom of Choice. Schreier graphs are used to prove the NielsenSchreier theorem in chapter 4. For later reference, we also classify Schreier graphs of (normal) subgroups of free groups. Chapter 5 starts with an analysis of the use of the Axiom of Choice in the proof of the NielsenSchreier theorem. Then we introduce representative functions  a tool for constructing choice functions from bases. They are used to deduce the finite Axiom of Choice from NielsenSchreier, and to prove the equivalence of a strong form of NielsenSchreier and the Axiom of Choice. Using FraenkelMostowski models, we show that NielsenSchreier cannot be deduced from the Boolean Prime Ideal Theorem. Chapter 6 explores properties of free abelian groups that are similar to those considered in chapter 5. However, the commutative setting requires new ideas and different proofs. Using representative functions, we deduce the Axiom of Choice for pairs from the abelian version of the NielsenSchreier theorem. This implication is shown to be strict by proving that it doesn't follow from the Boolean Prime Ideal Theorem. We end with a section on potential applications to vector spaces.
