It is a classical result of Dedekind that if A is a free abelian group, then so is any sub-group of A, and its rank is no more than that of A (see Chapter 6). Our purpose in this chapter and the next is to prove the non-commutative analogue, the celebrated Nielsen-Schreier theorem. In the non-abelian case, however, the rank of the subgroup may exceed the rank of the group (see Exercises 1.11, 1.15, 2.6, 2.9).

The two methods of proof given here, due to Nielsen (1921) and Schreier (1927), are ostensibly very different (but see the comments at the start of Chapter 3 and in the Guide to the Literature) and lay the foundations for different aspects of the subsequent development of the subject. Thus, for example, Nielsen's method leads naturally to the theory of automorphisms of free groups (where there is currently much active interest), while Schreier's method provides the key to finding presentations of subgroups (see Chapter 9).

Schreier's method will be described first for the following three reasons: it seems to exhibit a basis for the subgroup in a more explicit way, it includes the case of infinite rank, and it gives a precise formula for the rank of the subgroup in the finite case. Neilsen's method enjoys other advantages, and the restriction to the case of finite rank can be avoided.

The proof which follows is divided into six steps, and the constructions are illustrated by a concrete example at each stage.