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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
Knowledge is an encyclopedia,
669:. 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
627:(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.
1151:"More generally the direct product of any finite number of free abelian groups is free abelian" - it might (I'm unsure) be useful to give a link to mathematical induction here, to give an idea for how the result is reached... perhaps something like "More generally, the direct product of any finite number of free abelian groups can be shown to be free abelian through
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396:) 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. —
1220:, as in "Solomon Lefschetz and Irving Kaplansky have claimed that ...". My immediate reaction was what the MOS guidance says - that their statements are not reasonable or based in evidence. I think we can easily say "have said" here, unless Kaplansky gives a reason to doubt it. (I don't have access to Kaplansky to check, so I leave it to you.)
891:
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).
718:
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.
890:
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
836:
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
794:
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
1114:
This may be too technical an article to be a good candidate for DYK in any case, so I wasn't planning on nominating it and am not too concerned about their stronger sourcing requirements, as long as the sourcing is good enough for GA standards. I added some sources for the definition of free abelian
437:
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
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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
1131:
This is just a thought, and not a suggestion that I've thought through very deeply. It may be useful for "in the corresponding polynomial, or vice versa" to demonstrate how isomorphisms are generally verified. Because we can do both of these mappings, it is isomorphic - but the general point is not
717:
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,
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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
357:
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
336:
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
268:
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
1386:
Thanks so much. Re the induction thing: I just thought it would be a nice pointer - students will probably come across the article and wonder how that's done, but I'm sure they're able to recognize it, and it's not important to include. It's been a while since I taught algebra, but that was the
859:
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
698:
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
1174:"it is a quotient of the free abelian group over" - we link quotient in the lead but not here - do you think that would be helpful? It is linked in the "Rank" section, but "overlinking" in the body can be useful when terms are not yet fully understood by the reader
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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
1063:
Hello there. I will be taking a look at the article. In your comments, you said that someone with familiarity would be helpful - I took a few graduate classes on algebra, and from a broad look, the article seems appropriate for those who entering the subject
1193:"Again, this is a group invariant" - this seems somewhat informal. Leaving it at "This is a group invariant" is probably enough - it's important enough to say again, but whether we need to stress its importance is not clear ... to me (your opinion welcome)
799:, Knowledge 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
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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. --
316:
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...
1082:
I've been watching some of the work on the article for a few days, so I'm glad you nominated it for GA status. My comments below, which you are free to disagree with. As long as I can understand your thought process, I think it's all good.
750:
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.
1335:
I'll leave it here, and if I think of anything more, I'll add some comments. I think it's clearly at GA level, but I hope my feedback can improve the article, or if I'm off with my comments, I hope I can understand better your choices.
1331:
As an aside, I find the citations to specific exercises amusing. Not because they're a problem, but because this is perhaps the only field we can do that sort of thing :) Hungerford was what we used for some classes, solid book.
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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.
945:. 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). —
894:
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,
940:
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
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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).
1284:
Rewrote to use a simple conjunction ("must have multiplicities summing to zero, and meet certain additional constraints") rather than trying to compare the two kinds of constraints using "beyond". —
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The last few days were just a little more polishing before I finalized the nomination; most of my edits to this article were much earlier (beginning in 2013 after a couple smaller edits earlier). —
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Just to be clear, I don't think the uncited paragraphs in "Definition and examples" problematic: They are common knowledge for who we are writing to. That may be a problem for DYK,
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Verifiable: Yes, with why I don't find the uncited text problematic above; reliable sources; no original research that I can see; no copyvio (Earwig's picks up nothing worrying)
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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
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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. --
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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.
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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.
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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.
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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.
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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).
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Changed from "claimed" to "argue". The point of the wording was that this is really a statement of opinion rather than mathematical fact. —
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Ordinarily the terminology used would be no problem. But because of the warning earlier in the article in the "Terminology" section:
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really that there is just one map, right? Maybe "in the corresponding polynomial, and vice versa" would be helpful for illustration.
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1324:- I like where this is. You may also consider moving it down in the section to be closer to the information relating to lattices.
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1068:. I'll take a closer look and offer my comments below soon - and if you disagree with anything I say, feel free to say so.
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and I find proofs (or proof outlines) of difficult theorems something that spices up the articles, i.e. proofs makes them
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Is induction really necessary? Just take the disjoint union of the bases. Why do you need to do it one step at a time? —
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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
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I meant that meets the conditions stated earlier in the same paragraph, but I tightened the wording to remove this. —
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On images: They are good; illustrative; the rationales for use are clear; captions are useful. Regarding the image
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Isn't it true that "G is a free abelian group" is simply equivalent to "G is isomorphic to some direct sum of Zs"?
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Every free abelian group is just isomorphic to some (possibly infinite) direct sum of Zs? Mention this earlier!
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Thanks for the review! I'll try addressing your comments individually over the next few days as I find time. —
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Ok, I added an example here, and a not-very-technical explanation of why this mapping is an isomorphism. —
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except in the two special cases mentioned, since a free group on more than one generator is not abelian".
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Ok, I think I've responded to everything; please take another look. —
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The lead is well-written and gives a good overview for the subject.
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limited group, not the group of random, generally curious, readers.
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943:abelian group
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312:Definition???
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1314:Neutral: Yes
1300:
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1015:Instructions
922:
893:
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384:TakuyaMurata
382:). However,
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191:Mid-priority
190:
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116:Mid‑priority
94:WikiProjects
52:
43:please do so
31:
30:
26:
1029:transcluded
673:, not only
671:interesting
667:interesting
358:f.g. case?
274:JackSchmidt
166:Mathematics
157:mathematics
113:Mathematics
1413:Categories
1311:Broad: Yes
982:Authorship
968:GA toolbox
876:Definition
249:free group
37:under the
1041:Reviewer:
1005:Templates
996:Reviewing
961:GA Review
675:technical
247:is not a
1214:MOS:SAID
1054:contribs
1010:Criteria
927:D.Lazard
923:equipped
629:D.Lazard
394:contribs
339:Suitangi
84:GA-class
47:reassess
1218:claimed
884:editor
782:YohanN7
736:YohanN7
701:YohanN7
684:YohanN7
193:on the
1109:though
663:reader
90:scale.
54:Review
1031:from
319:linas
1396:talk
1392:Urve
1378:talk
1368:Urve
1357:talk
1342:talk
1338:Urve
1290:talk
1267:talk
1248:talk
1229:talk
1202:talk
1183:talk
1164:talk
1141:talk
1121:talk
1095:talk
1074:talk
1070:Urve
1048:talk
1044:Urve
951:talk
931:talk
866:talk
850:talk
809:talk
805:Ozob
803:).
786:talk
772:talk
768:Taku
757:talk
740:talk
724:talk
720:Ozob
705:talk
688:talk
652:talk
648:Taku
633:talk
623:and
421:talk
417:Taku
402:talk
388:talk
380:diff
364:talk
343:talk
323:talk
300:talk
296:Daqu
292:also
278:talk
259:talk
255:Daqu
243:"A
185:Mid
49:it.
1415::
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96::
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