Talk:Free group

Free abelian group

Question, is the reason that a free abelian group is not the same as an abelian free group is that in an free group ab ≠ ba, necessarily, but in an abelian free group, a+b=b+a? —Preceding unsigned comment added by 67.120.99.192 (talk • contribs) 22 August 2005

All free groups are not abelian, as the word aba^{-1}b^{-1} is not the neutral element. Anyhow the free groups are the free objects in the category of groups and hence the name free group makes sense. A free abelian group is a free object in the category of abelian groups. The meaning of free abelian group is NOT: a free group which is abelian. This is a very important subtle difference. HenrikRueping (talk) 09:12, 15 April 2009 (UTC)

Very nice

-- Very nice page. After spending an hour staring at Hungerford's definition, it's nice to see the clear underlying idea. Thanks. 134.243.218.123 23:11, 22 March 2006 (UTC)

Possible revision and expansion

Hi everyone!

I've been working on a possible revision and expansion of this article for a while. Parts of it are now in good enough shape that it might make sense to move them into the main article in some form. Let me know what you think:

• User:Jim.belk/Draft:Free group

Please be aware that when I'm writing a draft, I tend to rewrite as much text as possible, on the theory that we can use whichever version comes out better. Jim 20:24, 22 September 2007 (UTC)

I don't think that what you have prepared is any way acceptable as a substitute for the present page. It is amateur and short on any extra useful information. This is not the way WP articles are written. Mathsci 21:22, 22 September 2007 (UTC)

I'm really not proposing replacing this page with the draft. I wrote a bunch of stuff, and now I'm proposing moving the best of it to this page. Please take a look and see if you like any of it. :-) Jim 21:29, 22 September 2007 (UTC)

It's probably best to follow normal pratice and add little bits at a time. Don't remove anything: there's a lot that you haven't included in your version and some of it is not concise or encyclopedic. The references of Magnus et al and Serre are classics. There is incidentally the whole connection of the lower central series of the free group and free Lie algebras due to Magnus that has not yet been covered; also Haagerup-Gromov a-T-menability of the free group. So please just add extra details bit by bit, trying to be as concise and general as possible, and don't attempt to rewrite the whole article. Mathsci 21:45, 22 September 2007 (UTC)

All right, I'll just go ahead and post some things. Feel free to undo any revisions you dislike. Jim 22:29, 22 September 2007 (UTC)

Looks good. I just reinstated the fact about the naming of free groups and made a reference to Magnus' work on free Lie algebras. Your way of including references in the text makes it a little bit hard to edit the original. Please try to avoid this if you change any of the mathematics sections: just add references at the end. Mathsci 11:58, 23 September 2007 (UTC)

You have now started adding trivialities. The rank of a free group is the rank of its abelianisation as a free Abelian group. What you added is long-winded and mathematically obvious. It's exactly what is not wanted in this article. Why not read the chapter on commutators in Magnus' book? (This is the first case G/G'.) The problem with what you have added is exactly what was wrong with your draft article. Sorry. Mathsci 18:35, 23 September 2007 (UTC)

I'm sorry that you didn't like that section, but I think its contents would be non-obvious to someone who wasn't a group theorist. In any case, I'm happy to work on other articles instead. Jim 19:14, 23 September 2007 (UTC)

This is a little bit WP:POV, don't you think? If you look at the history of the page, you might get some idea of consensus and of the level of the article. But what's wrong with Ab(G)=G/G'? When I came across this article, I was surprised to find no mention of group actions on trees. Mathsci 19:51, 23 September 2007 (UTC)

the contents of the topology and geometry sections from Jim's draft, in more concise form perhaps, can help provide additional examples/perspectives. Mct mht 05:53, 29 September 2007 (UTC)

Inverses in Generating Set

The section on construction of the free groups from a set S says to suppose that for each element a in S, its inverse a^-1 is also contained in S. My understanding of the generating set is that it should be an arbitrary set, and a word in the elements of S can contain either positive or negative powers of those elements. Could we consider revision of that section? ATC2 (talk) 18:43, 29 June 2008 (UTC)

Yes, it's wrong. Feel free to rewrite it. Though I would stay away from powers - it's easier just to use x and x⁻¹. --Zundark (talk) 18:58, 29 June 2008 (UTC)

A way to correct it without drastic changes is just to distinguish the free generating set S = { a, b, c }, from the monoid generating set T = S ∪ S⁻¹ = { a, a⁻¹, b, b⁻¹, c, c⁻¹ }. In other words, the problem is not with the a⁻¹, but rather with using the letter S to denote two distinct sets. Basically, the free monoid on T is very easy to understand, and makes a good foundation for the free group construction (which is T modulo the rewrite rules a⋅a⁻¹ → 1, a⁻¹⋅a → 1, etc.). JackSchmidt (talk)

Done. Thanks for the suggestion.ATC2 (talk) 22:11, 1 July 2008 (UTC)

Cyclically reduced?

The article says „A word is called cyclically reduced, if its first and last letter are not inverse to each other. Every word is conjugated to exactly one cyclically reduced word.“ But in $F_{2}=\langle x,y,z\rangle$, the word $xyz$ is cyclically reduced according to the definition, and so is it’s conjugate $c_{x^{-1}}(xyz)=x^{-1}xyzx=yzx$. It seems that the second quoted sentence is wrong, or am I misunderstanding it? --Nomeata (talk) —Preceding undated comment added 08:36, 6 May 2009 (UTC).

Fixed. JackSchmidt (talk) 14:57, 6 May 2009 (UTC)

Unique epimorphism?

The article says "This can be formalized by the following universal property: given any function ƒ from S to a group G, there exists a unique homomorphism...." and that the "...homomorphisms F_S → G are in one-to-one correspondence with functions..." Doesn't uniqueness require the additional constraint that the homomorphism be surjective? Harold f (talk) 17:31, 3 June 2009 (UTC)

No, the uniqueness is due to the fact that F_S is generated by S. The homomorphism F_S → G needn't be surjective (and obviously can't be if |F_S| < |G|). --Zundark (talk) 17:51, 3 June 2009 (UTC)

Tarski's Problem

There was some hubbub about Kharlampovich & Myasnikov's original independent proof not working correctly. I've heard they ultimately fixed it by employing a bunch of Sela's techniques, occasionally even employing his notation, which differed markedly form their own. It's apparently fairly clear their result should ultimately be regarded as an extension of Sela's work to cover decidability. It's however true that they had done a great deal of work before he publish his paper too, probably an awful lot of which has direct bearing on the decidability question which they ultimately answered. It's probably best to simply site them for the decidability without referring to their arguments around elementary equivalence. — Preceding unsigned comment added by 138.251.240.12 (talk) 13:20, 12 April 2012 (UTC)

Construction

The section on the construction as words contains true statements but lacks the important remarks that extra work is needed to prove that this is indeed a group. We need to prove, for example, that the group operation (concatenation followed by reduction) is well-defined, and (related) that the operation is associative. I'm not saying that we should do the proof, but we should mention that it's needed. — Preceding unsigned comment added by 65.115.83.195 (talk) 00:52, 20 July 2012 (UTC)