Every subgroup of a free group is free - Why is this a big deal to prove?

In every algebra book I’ve read, they say something like ‘the proof is nontrivial and requires covering spaces’ or something like that, and that’s fine. But they never bother to explain the reason it is nontrivial, and I’m not seeing it on my own.

It’s a big deal to prove because it is a big deal to prove. There is no known proof that doesn’t involve covering spaces, although Joe Rotman once published a proof that used only one dimensional complexes (that is, graphs). This argument then looks purely combinatorial. I once went through it in detail, but I no longer recall the details.

You simply cannot expect every simple statement, even true statement, to have a simple proof. Cf. Fermat.


Sure is hell isn’t mainstream sociology.

Trivial to understand, but non-trivial to put into the language of maths?

Well, the only reason it wouldn’t be nontrivial is if you see the ostensible outlines of a trivial proof. Do you have one in mind?

Well, it’s free if there’s no nontrivial relations between words, and if a subgroup has some relation, it should exist in the whole. But I’m missing something obvious, because I can’t see how it doesn’t just fall out of the definitions. The people who write the books see it, since they’re not talking about maximal trees and bouquets of circles just for the fun of it.

Don’t think of it as “a free group is one where there are no nontrivial relations between words”, because, in a sense, there are all kinds of nontrivial relations between words (e.g., in the free group on the generators {a, b}, we have three words “ab”, “a”, and “b”, and the nontrivial relation between them that the first is the product of the latter two). Rather, there are no nontrivial relations between the generators. That is, a free group is one where there’s some subset of its elements which you can think of as the “generators”, such that everything in the group can be built up from the generators, and there are no nontrivial relations between those generators.

However, a subgroup of such a group might not be simply of the form “Those elements which can be built up from this subset of the generators (aka, those words drawn from this sub-alphabet)”. For example, consider the free group on the generators {a, b}, and now move to the subgroup containing precisely those “words” in which the total number of "b"s and "b^{-1}"s is even.

To establish that this subgroup is free, we need to pick some subset of this subgroup which generates it, and show that there are no non-trivial relations between the elements of this subset. But it’s not immediately obvious how to do that, is it? Simply taking those generators of the original group which are still in the subgroup doesn’t suffice (since this is just {a}, which is not enough to generate all of the subgroup}; nor would taking the entirety of the subgroup as its generators work (since there will be all kinds of nontrivial relations around; e.g., “bbbb is the square of bb”, “ba is the inverse of a^{-1}b^{-1}”, and so on). A little thinking will solve the problem in this case, but it shows why it’s nontrivial in general.

Just to be absolutely clear, by “everything in the group can be built up from the generators”, I mean “everything in the group can be be built up from the generators using only the group operations (identity element, product, inverse)”, and by “and there are no nontrivial relations between those generators”, I mean there are no valid equations between terms built up in such a fashion from the generators other than those implied by the group laws.

I did mean to say generators, but yeah, I can see where it gets hairy now.

I think perhaps a “WARNING! MATHS!” addendum in the title might be in order…

Fine, but then the “Rational group” thread needs to have a “WARNING! NOT MATH!” addendum. Look at poor Chronos in post #17

I’m pretty sure he’s just being either silly or facetious there.

I know. So am I. :slight_smile:

Yes, it is true that, although rare, the mathematically minded can have a sense of humor :slight_smile:

Disclosure: I am a graduate student in statistics, the field of choice for people who don’t have the social skills to be accountants :slight_smile:

Further disclosure: I have a undergraduate minor in accounting.