26

The Nielsen–Schreier theorem states (in part):

Let $F$ be a free group, and $H\le F$ be any subgroup. Then $H$ is isomorphic to a free group.

I have seen the topological proof of this theorem using the correspondence between coverings and subgroups of the fundamental group. This has always struck me as using rather strong theory for what it is being used to prove (though I do appreciate the beauty of the argument).

In my head, I see the following (loose) argument:

Let $H$ be a subgroup of $F$, and assume that $H$ is not free. Then there exists some nontrivial relation $h_1h_2\dots h_n = 1$. But then this is also a nontrivial relation in $F$ implying that $F$ is not free, which is absurd. Thus $H$ is free.

Clearly, there must be some problem with this. What are the stumbling blocks here? An issue I see is that the exact notation a relation has always seemed a little vague to me (some reduced word equal to the identity?), but that doesn't seem like it ought to be a large enough problem to invalidate the argument.

Santana Afton
  • 6,978
  • 2
    There is a non-trivial relation in any group, free or not. Given any two elements $x$ and $y$, $xy=z$ for some $z$. – David C. Ullrich Jan 21 '18 at 19:51
  • This is a nice question, and honestly one I usually take for granted since I knew about this proof before knowing any of the word manipulation (combinatorics) used in dealing with free groups +1 – Andres Mejia Apr 07 '18 at 21:23
  • One reason to be suspicious of your argument is that it doesn't seem to use anything particularly about groups. Why wouldn't a similar argument show that a subsemigroup of a free semigroup is a free semigroup, or a sublattice of a free lattice is a free lattice? – bof Jul 02 '18 at 04:59
  • By the way, why do you say "isomorphic to a free group"? Surely anything that's isomorphic to a free group is a free group? – bof Jul 02 '18 at 05:00

1 Answers1

49

But then this is also a nontrivial relation in $F$

This step is not obvious at all. To get a nontrivial relation in $F$, you need to know that when you expand $h_1,\dots,h_n$ in terms of the free generators of $F$, then $h_1\dots h_n$ reduces to a nontrivial reduced word. Why should that be true?

Indeed, it may not be true. You have glossed over the fact that you need to choose some specific subset of $H$ which will be your free generators. If you choose incorrectly, then there will be nontrivial relations. For instance, suppose $F$ is free on generators $a$ and $b$, and you take $H$ to be the subgroup generated by $h_1=ab$, $h_2=aba$, $h_3=bab$. Then $h_1^3h_3^{-1}h_2^{-1}=1$ is a nontrivial relation among these generators of $H$ (but when you expand this out in terms of the generators of $F$, it reduces to the trivial word, so this does not contradict freeness of $F$). So $H$ is not freely generated by $\{h_1,h_2,h_3\}$. To prove $H$ is freely generated, you have to somehow come up with a special set of generators for which there will be no nontrivial relations.

Now for this particular $H$, that is not so hard (in fact, $H$ is all of $F$, so you can take $a$ and $b$ as your free generators). But if you have some completely arbitrary subgroup of $F$, it is not at all obvious how you would come up with a generating set such that any relation between them would still be a nontrivial relation in terms of the free generators of $F$, thus giving a contradiction as you suggest.

Eric Wofsey
  • 342,377
  • 3
    Excellent, thank you for the explanation! I appreciate the example. – Santana Afton Jan 21 '18 at 19:36
  • 9
    This answer is absolutely correct (+1), but it might be useful for the OP if I shift the emphasis. The main error in the proposed proof occurs at "Then there exists some nontrivial relation $h_1h_2\dots h_n=1$." That sentence presupposes that some $h_1,h_2,\dots,h_n$ are under discussion. But no such $h$'s have been introduced, so the sentence makes no sense. (In fact, as the answer explains, choosing appropriate $h$'s is a key ingredient of the proof.) – Andreas Blass Jan 21 '18 at 23:18
  • 1
    @AndreasBlass I interpret that sentence as being the place where you introduce the $h$s, but then the problem is that a "nontrivial relation" isn't defined (because we did not talk about a set of generators) or it's a wrong claim (as explained in this answer). – JiK Jan 22 '18 at 13:06