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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)
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)
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))
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)
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)
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)
To editor David Eppstein: 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, 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)
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Reviewing |
Reviewer: Urve ( talk · contribs) 04:31, 2 January 2022 (UTC)
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
one level below. 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.
Urve (
talk) 04:31, 2 January 2022 (UTC)
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.
For the formalities,
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.
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. Urve ( talk) 05:34, 2 January 2022 (UTC)
Free abelian group has been listed as one of the
Mathematics good articles under the
good article criteria. If you can improve it further,
please do so. If it no longer meets these criteria, you can
reassess it. Review: January 3, 2022. ( Reviewed version). |
This article is rated GA-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
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)
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)
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))
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)
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)
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)
To editor David Eppstein: 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, 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)
GA toolbox |
---|
Reviewing |
Reviewer: Urve ( talk · contribs) 04:31, 2 January 2022 (UTC)
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
one level below. 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.
Urve (
talk) 04:31, 2 January 2022 (UTC)
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.
For the formalities,
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.
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. Urve ( talk) 05:34, 2 January 2022 (UTC)