The idea behind Nielsen's original proof of the subgroup theorem is as follows. Given a finite subset U of a free group F, perform certain operations on U (called Nielsen transformations) that reduce it to a set V of a special form (called Nielsen reduced). Since it can be shown that <U> = <V> and that V is a basis for <V>, the subgroup theorem follows in the finitely-generated case. That <U> = <V> is obvious, and the freeness of <V> follows from Proposition 1.3 in much the same way as the freeness of <B> in Chapter 2. Another point in common with Schreier's method is the crucial role played by a wellordering of F, which is used here to prove that any finite U can be carried by Ntransformations into an N-reduced V (Theorem 1).

As in the preceding chapter, we illustrate the steps in the proof by giving a specific example, and then go on to derive a series of consequences of the method, including its extension to cover the non-finitely-generated case.

The finitely-generated case

Let F = F(X) be a free group and U a finite subset of F. As the words in U together involve only a finite set of generators from X, we can assume (by Exercise 1.10) that X is finite. We think of U = (u1, u2, …, un) as an ordered set and, as usual, write l(w) for the length of w ∈ F as a reduced word in X±.