(Nielsen–Schreier theorem)
Every subgroup $H$ of a discrete free group $G$ is itself a free group. Moreover, if $G$ is free on $k$ generators and $H$ has index $n$ in $G$, then $H$ is free on $n k - n + 1$ generators.
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 $G = F(S)$ is, by the van Kampen theorem, a fundamental group of a bouquet^{1} of $S$ many circles, which is in particular a connected 1-dimensional CW-complex $X$ (in simpler language, a connected graph). By general covering space theory, given a (pointed) connected space $X$ with $\pi_1(X, x) = G$, subgroups $H \subseteq G$ are in bijective correspondence with isomorphism classes of connected covering spaces $p: (E, e) \to (X, x)$, with $\pi_1(E, e) \cong H$. Now, a covering space $E$ of a connected graph $X$ 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 $\pi_1(E, e)$ is a free group.
The second statement is proved by observing that the Euler characteristic of $X$ is $\chi(X) = 1 - \rho(G)$, where $\rho(G)$ is the number of generators of the free group $G$, and $\chi(E) = n\chi(X)$ if $p \colon E \to X$ is a covering space with $n$ points in each fiber.
Full details may be found in May. Key technical ingredients include: (1) each connected graph $E$ contains a maximal tree $T$ (using Zorn's lemma), (2) the quotient map $E \to E/T$ is a homotopy equivalence, and $E/T$ 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 $X \sslash G$ formed from a $G$-set $X$, 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 $F(S)$ 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 $S$. This is the free groupoid $*\sslash F(S)$ 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 $\mathbf{E}(F(S))$, the action groupoid $F(S)\sslash F(S)$ of $F(S)$ acting on itself from the right and that its quotient by the $F(S)$-action from the left recovers the original groupoid with fundamental group $F(S)$. If we instead quotient only by the given subgroup $H \hookrightarrow F(S)$, we obtain a connected groupoid with fundamental group $H$ (meaning simply that at any point or object $x$ of the groupoid, the group $\hom(x, x)$ is isomorphic to $H$).
Thus, consider the quotient $H \backslash \mathbf{E}F(S) \simeq H \backslash \backslash *$, the connected groupoid whose fundamental group is $\pi_1 = H$. By the properties of free groupoids discussed at free groupoid – fundamental group it is sufficient to show that $H \backslash \mathbf{E}F(S)$ is isomorphic to the free groupoid on a connected directed graph. But $\mathbf{E}F(S)$ itself is the free groupoid on a directed graph, namely on the action graph $F(S)\sslash S$ (which is the same as the Cayley graph? given by a set $S$ of generators and no relations), and we may consider the quotient graph $H \backslash (F(S) \sslash S)$. The free groupoid functor $F$ preserves this quotient. Thus we have
as desired.
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.
The above simple homotopy-theoretic proof was indicated in (Higgins), also (Gilbert-Porter).
See at pid - Structure theory of modules for details.
The restricted statement that every subgroup of a free abelian group is itself free was originally given by Richard Dedekind.
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
A bouquet of circles is the coproduct of a collection of circles, each with a basepoint, in the category of pointed spaces. In other words, a wide pushout in Top of inclusions of a 1-point space into circles, with the evident pointing. ↩