This has a number of different proofs. The first proof, perhaps nowadays the most familiar proof, is based on covering space theory (in particular, covering spaces of topological graphs). The second proof is quite similar in spirit but is based on groupoids (another way of viewing homotopy 1-types), and has the advantage that it circumvents the point-set topology considerations inherent to covering space theory.
of theorem 1
A free group is, by the van Kampen theorem, a fundamental group of a bouquet1 of many circles, which is in particular a connected 1-dimensional CW-complex (in simpler language, a connected graph). By general covering space theory, given a (pointed) connected space with , subgroups are in bijective correspondence with isomorphism classes of connected covering spaces , with . Now, a covering space of a connected graph is also a connected graph. But any connected graph is homotopy equivalent to a bouquet of circles, whose fundamental group is a free group. Thus is a free group.
The second statement is proved by observing that the Euler characteristic of is , where is the number of generators of the free group , and if is a covering space with points in each fiber.
Full details may be found in May. Key technical ingredients include: (1) each connected graph contains a maximal tree (using Zorn's lemma), (2) the quotient map is a homotopy equivalence, and is a bouquet of circles.
This topological proof can be reformulated in more algebraic language, using a little groupoid theory (groupoids being homotopy 1-types). A key construction here is the action groupoid formed from a -set , also called a homotopy quotient or homotopy orbit space.
of theorem 1
By the discussion at free groupoid – fundamental group, we may think of the free group as the fundamental group of a homotopy 1-type which is freely built from a single vertex and one loop from that vertex to itself for each element in . This is the free groupoid on this directed graph (a bouquet of circles). It is a classical fact (see at universal principal bundle) that the universal cover of this is the contractible groupoid , the action groupoid of acting on itself from the right and that its quotient by the -action from the left recovers the original groupoid with fundamental group . If we instead quotient only by the given subgroup , we obtain a connected groupoid with fundamental group (meaning simply that at any point or object of the groupoid, the group is isomorphic to ).
Thus, consider the quotient , the connected groupoid whose fundamental group is . By the properties of free groupoids discussed at free groupoid – fundamental group it is sufficient to show that is isomorphic to the free groupoid on a connected directed graph. But itself is the free groupoid on a directed graph, namely on the action graph (which is the same as the Cayley graph? given by a set of generators and no relations), and we may consider the quotient graph . The free groupoid functor preserves this quotient. Thus we have
The original algebraic proof of theorem 1 was rather long and complicated, based on Nielsen’s method of short cancellations in combinatorial group theory based on words (Nielsen’s transformations of words, word length functions, …). The class of ‘projective groups’ ( that is, in this context, projective objects in the category of groups) coincides with the class of free groups.
Every subgroup of a free abelian group is itself a free abelian group.
See at pid - Structure theory of modules for details.
Jakob Nielsen proved the statement for finitely-generated subgroups in 1921. The full theorem was proven in
The topological proof is due to
and another one due to Jean-Pierre Serre.
Versions of the topological proof are given in many places. One is
Another can be found reviewed towards the end of
Other related texts include
H. Zieschang, Über die Nielsensche Kürzungsmethode in freien Produkten mit Amalgam, Invent. Math. 10, 4–37, 1970.
W. Magnus, A. Karras, D. Solitar, Combinatorial group theory
R. Lyndon, P. Schupp, Combinatorial group theory, Springer 1977 (Russian transl. Mir, Moskva 1980)
A groupoid-based proof of the Nielsen-Schreier theorem appears as theorem 9 in chapter 14 in
Similar material can also be found in