Intereting Posts

Not empty omega limit set
For which cases with $2$ or $3$ prime factors do formulas for $gnu(n)$ exist?
Probability of winning the game 1-2-3-4-5-6-7-8-9-10-J-Q-K
A layman's motivation for non-standard analysis and generalised limits
Will the Fermat's last theorem still hold if algebraic and transcendental numbers are introduced?
Intersecting maximal ideals of $k$ with $k$
threshold of n to satisfy $a^n <n^a$
What are some easier books for studying martingale?
How do I find the marginal probability density function of 2 continuous random variables?
Tensor product and Kronecker Product
First-order logic advantage over second-order logic
Polynomial $p(x) = 0$ for all $x$ implies coefficients of polynomial are zero
A constrained extremum problem
Closed form for $\sum^{\infty}_{{i=n}}ix^{i-1}$
Good Physical Demonstrations of Abstract Mathematics

There are two proofs of Nielsen-Schreier that I know of. The theorem states that every subgroup of a free group is free. The first proof uses topology and covering space theory and is rather elegant. The second uses combinatorial techniques on a free group of words with no relations.

Is there a more algebraic proof which somehow just uses the universal property of free groups and maybe other properties of groups that are proved more “algebraically”?

I’m interested because groups are defined purely algebraically by equations, and some proofs that a subgroup of a free abelian group is free abelian have a far more algebraic flavour. So perhaps there is some proof of Nielsen-Schreier that also has a more algebraic flavour?

- What can we say about the kernel of $\phi: F_n \rightarrow S_k$
- Commutator subgroup of rank-2 free group is not finitely generated.
- Why the group $< x,y\mid x^2=y^2>$ is not free?
- Finding subgroups of a free group with a specific index
- Prove that $PSL(2,\mathbb{Z})$ is free product of $C_2$ and $C_3$
- The free group $F_2$ contains $F_k$

Ideally I would like a proof that does not involve combinatorial properties of a group of words on generators; in other words preferably no facts from combinatorial group theory.

- $a \in G$ commutes with all its conjugates iff $a$ belongs to an abelian normal subgroup of $G$
- Probability that $xy = yx$ for random elements in a finite group
- Classifying groups of order 90.
- Can the semidirect product of two groups be abelian group?
- Direct products of infinite groups
- Does the intersection of two finite index subgroups have finite index?
- I would like to show that all reflections in a finite reflection group $W :=\langle t_1, \ldots , t_n\rangle$ are of the form $wt_iw^{-1}.$
- Adapting a proof on elements of order 2: from finite groups to infinite groups
- Equivalence of tensor reps & tensor products of reps
- Criterion for being a simple group

So the question can be marked off as answered…

I don’t think there can be a “purely functorial” proof that a subgroup of a free group is free (that is, a proof using just the universal property of the free group). If there were such a proof, one would naturally expect that it can be applied to work for the relatively free groups in any variety of groups. But the only varieties of groups in which subgroups of free groups are always free are the variety of all groups, the variety of all abelian groups, and the varieties of abelian groups of prime exponent. So this argues strongly against the existence of such a proof.

As long as I’m writing this as an answer, I’ll note that the technical name for such varieties is *Schreier varieties.* That is, a variety $\mathfrak{V}$ of algebras (in the sense of universal algebra) is said to be a *Schreier variety* if and only if every subalgebra of a free $\mathfrak{V}$-algebra is itself a free $\mathfrak{V}$-algebra. The proof that the only Schreier varieties of groups are the ones listed above is due to Peter Neumann and James Wiegold in *Schreier varieties of groups*, Math. Z. **85** (1964) 392-400. An alternate proof was given by Peter Neumann and Mike Newman, *On Schreier varieties of groups*, Math. Z. **98** (1967) 196-199.

A proof can also be found in Hanna Neumann’s book **Varieties of Groups**, in section 4.3.

Recently emerged a purely algebraic proof (author’s claim) employing diagram chasing of some of the wreath product’s **functorial** properties. By authors Ribes, L.; Steinberg, B. under the title: “A wreath product approach to classical subgroup theorems”. Enseign. Math. (2) 56 (2010), no. 1-2, 49–72., also available at the arXiv: Ribes, L.. They effectively use the universal property definition and are capable of proving the Kurosh’s Subgroup Theorem as well.

There is an algebraic version of the topological proof using **covering morphisms of groupoids**, due to Philip Higgins,

Higgins, P.~J. “Presentations of groupoids, with applications to groups”.

*Proc. Cambridge Philos. Soc.* 60 (1964) 7–20.

and the downloadable

Higgins, P.J. *Notes on categories and groupoids*, Mathematical Studies, Volume 32.

Van Nostrand Reinhold Co. London (1971); Reprints in Theory and Applications of

Categories, No. 7 (2005) pp 1–195.

He uses the solution of the word problem for free groupoids, though I think this can be avoided by using the (functorial) fact that if $p:G \to H$ is a covering morphism (actually fibration is sufficient) of groupoids, then the pullback functor $p^*: Gpds/H \to Gpds/G$ has a right adjoint and so preserves colimits. This result has other applications.

Addition 27 January, 2014: For the last paragraph, I can now refer you to my answer to this mathoverflow question. The point is that a subgroup $H$ of a group $G$ defines a covering morphism of groupoids $p:Tr(G,H) \to G$ where the first groupoid is the action groupoid of $G$ on the cosets of $H$. So the vertex groups of $Tr(G,H)$ are all isomorphic to $H$. You can then use the colimit preserving properties of $p_*$ as described in the answer to prove that if $G$ is a free group, then $Tr(G,H)$ is a free groupoid. It is also connected (= transitive). So choosing a maximal tree shows that any of its vertex groups are free groups. What is called a *Schreier transversal* is just a maximal tree in the groupoid $Tr(G,H)$.

- Prove $BA – A^2B^2 = I_n$.
- How to find non-isomorphic trees?
- What are the symmetries of a colored rubiks cube?
- If all of the integers from $1$ to $99999$ are written down in a list, how many zeros will have been used?
- natural problem where a Lebesgue integral turns out to be useful
- Cohomology group of free quotient.
- Is there a retraction of a non-orientable manifold to its boundary?
- Conceptual question about Locally Convex Spaces
- $\text{Evaluate:} \lim_{b \to 1^+} \int_1^b \frac{dx}{\sqrt{x(x-1)(b-x)}}$
- What's the largest possible volume of a taco, and how do I make one that big?
- Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree
- Show that $d'(x,y)=min${$1,d(x,y)$} induces the same topology as $d$
- Integral of $1/z$ over the unit circle
- Is $f(x)=\sup_{y\in K}g(x, y)$ a continuous function?
- Transcendental Extensions. $F(\alpha)$ isomorphic to $F(x)$