Talk:Free abelian group

More care needed with terminology
The article presents this theorem toward the end:

"Theorem: Let F be a free abelian group generated by the set and let  be a subgroup. Then G is free."

Ordinarily the terminology used would be no problem. But because of the warning earlier in the article in the "Terminology" section:

"Note that a free abelian group is not the same as a free group that is abelian; a free abelian group is not necessarily a free group. In fact the only free groups that are abelian are those having an empty basis (rank 0, giving the trivial group) or having just 1 element in the basis (rank 1, giving the infinite cyclic group). Other abelian groups are not "free groups" because in free groups ab must be different from ba if a and b are different elements of the basis."

the use of the word "free" in the theorem is likely to be quite confusing to beginners. I would strongly suggest using consistent terminology throughout the article.

Also, the warning given above could be worded so as to be less confusing: Just state the simpler fact:

"A free abelian group is not a free group except in the two special cases mentioned, since a free group on more than one generator is not abelian".

[Note: The software is not giving me any window in which to enter an edit summary.]Daqu (talk) 22:20, 11 February 2009 (UTC)


 * I fixed the statement of the theorem. Using the standard abbreviation of "free" for "free abelian" in the proof seems fine, but was a little confusing in the statement itself.  I'm not sure how to improve the terminology warning.  The article itself is somewhat boring and unbalanced.  Presumably when it is rebalanced the terminology section's content will be more carefully integrated into the article as a whole.  Right now it is just another trivia statement that could be included in the numbered list under Properties.
 * When you create a new section, the edit summary will just be the section heading, which is a good edit summary in most cases (including this one). JackSchmidt (talk) 22:44, 11 February 2009 (UTC)


 * Thanks for your comment. Normally I would not mind seeing "free" used to mean "free abelian" if the context makes this clear. But that struck me as unwise in an article that also uses "free" to mean just plain "free" (even emphasizing the distinction).Daqu (talk) 06:52, 16 February 2009 (UTC)

Definition???
This article, as it currently stands, lacks a formal definition. It jumps straight from 'example' to 'properties'. The lead has a single-sentence informal definition. I mean, I can guess, but it would be nice to have the explicit axioms... linas (talk) 16:44, 2 September 2012 (UTC)

Free-abelian groups
Although "free abelian groups" is a quite standard terminology for this kind of groups, it is misleading (not every free abelian group is free and abelian, in fact only exceptional cases are). I suggest naming them "free-abelian groups" which is much more precise ("free-abelianicity" is the true defining property of them) and supposes a very minimal change (in mnemotechnical terms). (Suitangi (talk) 08:18, 10 September 2012 (UTC))

Subgroup Closure
This proof is a horrible mess, and not very informative. In the finitely-generated case, it is easy (since a subgroup of a finitely generated free abelian group is finitely generated and torsion-free, and hence free abelian). Does anyone object to stating the complete result, but only proving the f.g. case? 71.227.119.236 (talk) 17:33, 2 April 2013 (UTC)


 * This section was tagged as too technical, and I agree. In order to handle it, I removed the overly formal "theorem-proof" part of the section, leaving only the statement of the result, and I added a sentence about the relation between this fact and the fact that the infinite cyclic group only has infinite cyclic subgroups (diff). However, disagreed, restored the section to the way it was previously (with the technical proof and without my new sentence), and removed the "technical" tag. Perhaps we could have some more discussion about what to do here? Because I don't think leaving it as it is works very well, but getting into an edit war without forming a consensus is also a bad idea. —David Eppstein (talk) 15:24, 28 November 2013 (UTC)


 * I restored the proof since (a) it is not an "original research", or least doesn't seem to be, and (b) generally speaking, removing materials to make things less technical is a bad ideal. A "non-topological proof" is always very messy, as I understand, and at least the proof shows how messy it can get. (Sorry, I didn't notice the new sentence, I'm restoring it right now.) -- Taku (talk) 16:22, 28 November 2013 (UTC)


 * I agree with IP user and David: As far as I know, this proof (in the case of non finitely generated groups) is useful only to prove the theorem and is not really of encyclopedic interest. A reference would suffice. On the other hand, surprisingly, the article does not mention the full theorem in the case of finitely generated groups: given a subgroup of a finitely generated free Abelian group, there is a basis $$e_1, \ldots, e_n$$ of the larger group and integers $$d_1|d_2|\cdots |d_k$$ (each integer divides the next one) such that $$d_1e_1, \ldots, d_ke_k$$ is a basis of the smaller group. Moreover Smith normal form, provides both a constructible proof of this theorem and a practical way to compute a basis of the subgroup. All of this is widely used in crystallography and computer algebra (linear algebra over the integers). Also recent results show that the computational complexity of this computation is of the same order as the complexity of the computation of a determinant. IMO, the proof in the non finitely generated case should be removed, and the proof in the finitely generated case should be replaced by expanding the above brief summary of the subject. D.Lazard (talk) 17:17, 28 November 2013 (UTC)

Ok. D. Lazard is very persuasive (as usual?). I like a newer version. While I don't think the proof put before was bad, it was not particularly good either, so I'm fine. -- Taku (talk) 00:37, 29 November 2013 (UTC)


 * I am an encyclopedia reader and I find proofs (or proof outlines) of difficult theorems something that spices up the articles, i.e. proofs makes them interesting. I could argue that this makes the presence of the full proof being of "encyclopedic interest". I'm trying here to say that the "not of encyclopedic interest"-argument means little, because it can be made to mean anything. Pure difficulty (technicality) of a proof doesn't render it (in my mind) as not being of interest. In the present case, finite versus infinite, makes it interesting, not only technical.


 * The formerly present proof is admittedly messy (all the inline LaTex), but it could surely be fixed and transformed into a fairly short proof outline. YohanN7 (talk) 16:44, 29 November 2013 (UTC)


 * For the same reasons as stated above, a proof of the finite case (though simple) would not be of "encyclopedic interest" because it isn't interesting by any measure - except for those active or learning in the field. True, those would be encyclopedia readers too, but belonging to an inherently limited group, not the group of random, generally curious, readers.YohanN7 (talk) 18:37, 29 November 2013 (UTC)


 * Actually, I think the finite case is much more interesting. I think it's the part of the proof where all the interesting ideas occur.  Amplifying the argument to the infinite case feels to me like a rather standard application of Zorn's lemma: Define a structure that represents a partial solution, order those structures, use Zorn's lemma to find a maximal element, and prove that a maximal element must be the entire object.  The only step that is not exactly the same as in all other Zorn's lemma arguments is the last, and all of the insight needed to do that step already occurs in the finite case.  Ozob (talk) 03:11, 30 November 2013 (UTC)
 * See, there is a skeleton of a proof outline. YohanN7 (talk) 10:00, 30 November 2013 (UTC)
 * Lang (who we are citing for the Zorn-based proof) seems to agree with you. He gives the finitely generated proof in the main text and then says that the general case (in an appendix) is basically the same thing using transfinite induction. —David Eppstein (talk) 03:53, 30 November 2013 (UTC)
 * But a "standard application" is of encyclopedic nature since, well, "standard". We shouldn't say "this is standard/usual so we skip it" since that kind of defeats the purpose of this encyclopedia. It is important not to hide the technical nature. -- Taku (talk) 04:26, 30 November 2013 (UTC)
 * I blatantly agree. YohanN7 (talk) 10:00, 30 November 2013 (UTC)
 * To me, calling something "standard" suggests that it is less interesting, not more. It says that someone who understands the ideas involved in similar proofs will see how to arrange the same ideas to produce a proof of the result of interest.  Of the four steps in my proof sketch above, I would call the first three standard.  Someone who has experience with Zorn's lemma arguments will find those steps familiar, straightforward, maybe even trivial.  The relevant details might be interesting to someone who is unfamiliar or unpracticed with Zorn's lemma, but because they are standard, they are not of interest in an article that is not about Zorn's lemma.  Since Wikipedia is an encyclopedia, not a textbook, Wikipedia should not include parts of arguments that do not improve the reader's insight, and that means that it should not include arguments that are standard (except in an article about a standard argument which is of encyclopedic interest, like the pigeonhole principle).  Ozob (talk) 05:00, 1 December 2013 (UTC)

Every free abelian group is just isomorphic to some (possibly infinite) direct sum of Zs? Mention this earlier!
Isn't it true that "G is a free abelian group" is simply equivalent to "G is isomorphic to some direct sum of Zs"?

It seems true, due to the fact of having a basis. Yet I can't quite tell from the article. If it is true, this should be mentioned early and often, in my opinion in the very first sentence of the article! It's preposterous that it wouldn't be mentioned earlier, making readers think this is a far more difficult concept than it really is.

Wolfram MathWorld's page here: http://mathworld.wolfram.com/FreeAbelianGroup.html seems to mention this up front except it says direct product instead of direct sum, which is not the same, and seems not true, so now I'm a bit confused about their page. Cstanford.math (talk) 00:33, 4 September 2018 (UTC)
 * It is true, and stated explicitly in the "Direct sums, direct products, and trivial group" section. It's not in the lead, because the relevant part of the lead (its third paragraph) is oriented around what the elements are, not what the whole group is, and if you expanded what the elements of a direct sum are you would get something more or less the same as what's already there. —David Eppstein (talk) 01:27, 4 September 2018 (UTC)

Definition
Thank you for trying to improve this article, in particular by adding a definition. Unfortunately, the definition that you added is incorrect in the sense that it differs from the usual one. In short, you define a free abelian group essentially as the pair of an abelian group and a basis of it, while the standard definition is to be an abelian group such that a basis exists. The main drawback of your definition is that, with it, if you change of basis, you change the abelian group. So, the assertion that a free abelian group may have more than one basis becomes wrong (by the way, a free abelian group has always more than one basis).

Also, please, avoid also to define algebraic structures as tuples: if you define an (abelian) group as a pair of a set and an operation, then many common formulations and formulas become incorrect (for example, $$x\in G$$ would be formally wrong, as a pair is not a set). The standard (and less technical) way is to define a group as a set equipped with an operation. This makes also easier to consider several structures on the same set, such as in the case of a ring (a ring is an abelian group under addition and a monoid under multiplication). D.Lazard (talk) 10:41, 28 December 2021 (UTC)
 * What does "equipped with" mean, formally, except that you have a tuple specifying how it is equipped? I note that the pair (set,operation) appear prominently in the first paragraph of the definition section of abelian group. I take your point about existence of a basis vs specification of a basis, though (much like the usual distinction between vector spaces specified abstractly vs specified as coefficients for a specific basis). —David Eppstein (talk) 16:11, 28 December 2021 (UTC)