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for ease of linking I gave fundamental theorem of finitely generated abelian groups a stubby entry of its own.
In the course of this I also created a stub for structure theorem for finitely generated modules over a principal ideal domain which used to be requested at principal ideal domain.
I have added some content to the entry. Right now it reads like so:
fundamental theorem of finitely generated abelian groups:
Every finitely generated abelian group $A$ is isomorphic to a direct sum of p-primary cyclic groups $\mathbb{Z}/p^k \mathbb{Z}$ (for $p$ a prime number and $k$ a natural number ) and copies of the infinite cyclic group $\mathbb{Z}$:
$A \;\simeq\; \mathbb{Z}^n \oplus \underset{i}{\bigoplus} \mathbb{Z}/p_i^{k_i} \mathbb{Z} \,.$The summands of the form $\mathbb{Z}/p^k \mathbb{Z}$ are also called the p-primary components of $A$. Notice that the $p_i$ need not all be distinct.
fundamental theorem of finite abelian groups:
In particular every finite abelian group is of this form for $n = 0$, hence is a direct sum of cyclic groups.
fundamental theorem of cyclic groups:
In particular every cyclic group $\mathbb{Z}/n\mathbb{Z}$ is a direct sum of cyclic groups of the form
$\mathbb{Z}/n\mathbb{Z} \simeq \underset{i}{\bigoplus} \mathbb{Z}/ p_i^{k_i} \mathbb{Z}$where all the $p_i$ are distinct and $k_i$ is the maximal power of the prime factor $p_i$ in the prime decomposition of $n$.
Specifically, for each natural number $d$ dividing $n$ it contains $\mathbb{Z}/d\mathbb{Z}$ as the subgroup generated by $n/d \in \mathbb{Z}\to \mathbb{Z}/n\mathbb{Z}$. In fact the lattice of subgroups of $\mathbb{Z}/n\mathbb{Z}$ is the formal dual of the lattice of natural numbers $\leq n$ ordered by inclusion.
Example 1
For $p$ a prime number, there are, up to isomorphism, two abelian groups of order $p^2$, namely
$(\mathbb{Z}/p\mathbb{Z}) \otimes (\mathbb{Z}/p\mathbb{Z})$and
$\mathbb{Z}/p^2 \mathbb{Z} \,.$Example 2:
For $p_1$ and $p_2$ two distinct prime numbers, $p_1 \neq p_2$, then there is, up to isomorphism, precisely one abelian group of order $p_1 p_2$, namely
$\mathbb{Z}/p_1 \mathbb{Z} \oplus \mathbb{Z}/p_2 \mathbb{Z} \,.$This is equivalently the cyclic group
$\mathbb{Z}/p_1 p_2 \mathbb{Z} \simeq \mathbb{Z}/p_1 \mathbb{Z} \oplus \mathbb{Z}/p_2 \mathbb{Z} \,.$The isomorphism is given by sending $1$ to $(p_2,p_1)$.
Example 3:
For $n = 12 = 2^2 \cdot 3$ then
$\mathbb{Z}/12 \mathbb{Z} \simeq \mathbb{Z}/2^2 \mathbb{Z} \oplus \mathbb{Z}/3\mathbb{Z} \,.$How about that Invariant factor decomposition where an order divides the successor order?
I have to admit I never understood the purpose of this decomposition.
Probably there should be a remark on that in the entry, too. Myself, for the moment I need to focus my energy of the primary decomposition.
Urs, I don’t want to seem rude, but why the spelling out of such details in the examples? Particularly example 3 feels superfluous or redundant.
Todd, what’s going on. In don’t understand your reactions as of late. These are the archetypical examples illustrating the theorem. Not giving an example just because it is trivial to the expert runs counter the idea of writing exposition in the first place. And what would be the harm, anyway? I am preparing this for people who appreciate these examples, and I trust that those readers who don’t need to see them lose no sleep by just scrolling past.
I have added a section (here) on the graphical representation of the $p$-primary decomposition as used in the context of the Adams spectral sequence.
In don’t understand your reactions as of late.
I don’t understand what actions of late you’re referring to. But you should interpret my actions as not being hostile.
I am preparing this for people who appreciate these examples
That may answer my question. Are they students you are currently teaching?
Yes, I am compiling this for Introduction to Stable homotopy theory – 2.
Urs, I have just finished writing a review of a paper by Heuts and Lurie, on Ambidexterity. They are notes on some lectures in a minicourse that Lurie gave and contain some nice pages on abelian groups and spectra that might be useful to your students. They were in Contemporary Maths vol. 613.
Okay, thank you Urs for explaining. And please accept my apologies for indeed coming off as rude in comment #5.
I hope you don’t mind: I added a sentence that the following examples may be useful for instructional purposes, and also replaced a $\otimes$ in example 1 by a $\oplus$.
Todd, thanks for catching that typo. And thanks for explaining yourself, too.
I am fine with your addition, but I am still puzzled by your attitude here.
Isn’t an example always “for illustrative or instructional purposes”. What are you worried about? That people see a statement on the $n$Lab too trivial for them to stand? Is there a worry that the $n$Lab is not sophisticated enough?
I am thinking that, on he contrary, the $n$Lab could use loads more additions that serve “illustrative or instructional purposes”. For example most of our entries on category theory are lousy as a source for anyone turning to them who is not already half-way an expert. They could do with plenty of illustrative and instructional examples, and I wouldn’t think that each of them would need to be prefaced by an apology.
You and me we had run into a similar situation recently: I had suggested that the entry Cartesian product needed more examples. I realized this when I linked to it from an instructional site and noticed that the link was pretty useless for the kind of people expected to follow it.
Your first reaction was to suggest that it would be better without. In the end somehow I convinced you, and then you added a beautiful long list of examples. I think this addition pushed this particlar entry from something that makes a neglected appearance to anyone passing by to the level of becoming a valuable resource of sorts.
I think something similar should eventually to be done to many $n$Lab entries.
For an entry that starts out being about some sophisticated topic, it is okay to assume that the reader won’t be hoping to find instruction and illustration. But for entries that by design are about basic concepts, then if we have them at all (and it is good to have them), it is not a shame but a virtue for them to offer basic instruction and illustration.
For an entry that starts out being about some sophisticated topic, it is okay to assume that the reader won’t be hoping to find instruction and illustration. But for entries that by design are about basic concepts, then if we have them at all (and it is good to have them), it is not a shame but a virtue for them to offer basic instruction and illustration.
Actually, I disagree. Or at least I very much disagree with the first sentence I quoted, and somewhat with the second.
Isolated as I am, it’s usually it’s a big struggle for me trying to read the more advanced articles, and I’m sure the same is true for many readers. An example might be differential cohesive $(\infty, 1)$-topos. I’d really like to understand better this wonderful body of theory due largely to you, but I think I, and articles at that level, generally suffer from the lack of guiding examples right within the article to help guide the reader along. In my opinion, such examples are vital and they should come near the top of the article.
I don’t want to pick on that particular article much (under Examples I do see a link), but it seems to be a general observation, not just from me, that generally the nLab could do with a lot more examples, and especially for the more advanced concepts. For stuff that isn’t well known, and certainly not abundantly in the literature, people depend on the nLab as a primary source of information.
For less advanced subjects like the page on the structure theorem for finitely generated abelian groups: examples are certainly okay, but here there are a million places to learn about that stuff, both online and in textbooks. I don’t see examples for relatively elementary topics like that as being so critical to include in the nLab.
I am thinking very much of likely or typical readers of the nLab. If they include undergraduates at all, they will be very advanced undergraduates. Mostly they will be people like me who have a fair amount of training already, at a post-PhD level. But who turn to the nLab for guidance on more advanced topics, not always easy to find elsewhere gathered in one place. At least, that is my impression. We need the right kinds of (yes, illustrative, instructional) examples for those folks.
On an aesthetic level, I did find mention of very elementary examples just slightly jarring, mainly thinking of the general audience that I envision, who are not undergraduates in need of such classroom examples. I don’t want to make a big deal of that or this article, but generally speaking I find a lot of unevenness in level of treatment, where elementary details may be overdone and harder examples underdone. That’s of course bound to happen to some extent, but it’s a general observation I’d like to bear in mind as we move forward. For readers who may be surprised at what they perceive as that unevenness, I thought to include a little note that hints that the author may have included more mundane examples to help his students.
Well, I hope you get the idea.
I like examples 1 and 2, but example 3 seemed mismatched to them, so I tried to enhance it a little to bring it more in line.
I also don’t think anything useful is conveyed by the sentence “The following examples may be useful for illustrative or instructional purposes”, though I left it in for now. I hear what you’re saying, Todd, about unevenness, but it’s just a fact about how the nLab is written, with no top-down guidance. Are you worried that the presence of elementary examples in some places will lead people to be more annoyed or dissatisfied with their absence in other places?
Todd, thanks for this.
So that ambitious young reader who works through differential cohesive $\infty$-toposes, then turns to the applications discussed at Equivariant+cohomology+of+M2/M5-branes (schreiber), there learns that the equivariant Adams spectral sequence plays a role, but then doesn’t know (because no published source ever bothers to spend a word on it) what is meant by the $p$-primary part of the stable stems and how those pictures of the Adams stable page that people draw is supposed to encode them. He realizes that it must be so basic that nobody cares to explain, but on the other hand he just doesn’t see it on the spot. To his relief, the $n$Lab offers a link not just to the overarching $\infty$-theory, but also to all these little basic facts, just so that it’s all in one place. So he follows the entry to the fundamental theorem of finite abelian groups, gets the idea after a glance, and thus can happily resume his less basic pursuits.
Mike has now continued the list of examples.
I can see that the sentence I added is not terribly useful, but let me return to that towards the end of the comment.
I like your enhancement, Mike. In fact I don’t need to say anything more about this particular article; it seems all right now. Thank you!
Are you worried that the presence of elementary examples in some places will lead people to be more annoyed or dissatisfied with their absence in other places?
That could be a reaction, but I’m not sure. Sometimes when say a seminar speaker spends time giving examples of pretty simple things that everyone knows, and relatively less time illustrating harder material, people may be annoyed. But the situation is not exactly the same here, so I’m not sure.
Again, apologies for my ham-handed #5.
But I guess I also wonder a little bit about using the nLab as if for lecture notes for a course (which is the thought behind my clunky and useless sentence). I gather that’s partly how Urs is using it here.
Lecture notes can be great! And the earlier example 3 could fit in quite organically in a classroom setting, especially for undergraduates; the enhanced version seems more appropriate for the nLab. But when it comes to writing material whose purpose is to support classroom lectures, I wonder whether it might be a good idea to have some sort of notice to that effect. I remember thinking similar thoughts when someone that Urs knows had begun a similar classroom project, and Urs informed me privately what was going on.
(Somewhat related: in the early days of experimenting with the nLab, there were attempts at pedagogy which involved running conversations between John Baez or Urs talking with say Eric Forgy. To my eyes, some of those attempts now seem a little clunky or out of place. Maybe that’s just my personal taste. Anyway, some sort of advisory alerting the reader to the nature of such articles seems very appropriate, and indeed I think some of those articles did carry such labels, although I’d have to check.)
Hope this is making sense…
I agree in general that something appropriate for lecture notes may not be as appropriate for the main nLab. I haven’t noticed this being much of a problem, but it’s something to keep in mind.
I still find these thoughts puzzling. That the $n$Lab is in large parts lacking good exposition and introduction is one thing. But that this is regarded not as a deficiency but as a virtue or even a must, I find perplexing. If more of you regulars made it a habit to put your expositional material into $n$Lab entries instead of into old-style pdf-s (say Mike’s recent exposition of categorical logic), just imagine how immensely more valuable the $n$Lab would eventually become as a resource. How do you think young people are going to get to appreciate the sophisticated cutting edge stuff on the $n$Lab if there is nothing that leads them there? How could, in the absence of any constraints on space, additional exposition ever be regarded a drawback?
I was delighted when recently one professor decided to put his notes on Lie groupoids into the $n$Lab, and already then I didn’t understand why you, Todd, were being so sceptical. The main deficiency that I saw with these additions was that they ended up being minor.
My two pennies worth (after Brexit, much the same as two cents), I did actually react to that original example 3 as Todd did, surprised that something that elementary was appearing. But I didn’t see it as a problem as that page was not out of kilter with itself. In any case, it prompted Mike’s additions which make things just about right now for me. (Now you even have the means to answer a problem like: Find an $n$ for which there are 1 million distinct abelian groups of that order.) And the connection to sophisticated material such as stable homotopy groups gives the topic more salience.
Mind you, I say this is simple material, but as above (#3) I still don’t even know why people think the invariant factor decomposition is a good way to decompose abelian groups. What do you gain with $(\mathbb{Z}/2\mathbb{Z}) \oplus (\mathbb{Z}/4\mathbb{Z}) \oplus (\mathbb{Z}/3\mathbb{Z}) \oplus (\mathbb{Z}/9\mathbb{Z})$ in grouping it as $(\mathbb{Z}/6\mathbb{Z}) \oplus (\mathbb{Z}/36\mathbb{Z})$? I guess you see the highest order easily.
In general, as has been mentioned, the greatest problem with many nLab pages is lack of motivation and illuminating examples. I remember Cauchy complete category being vastly improved for me by the addition of the worked examples. Something like mate still leaves me unsatisfied. Why ever did someone think that worth defining? Is there not a motivating idea to include?
I’m for as much in the style of motivation for sheaves, cohomology and higher stacks as possible, but agree some of the early attempts at motivation, such as Understanding M-Set, could be lost, and all query boxes should be removed. And yes to the inclusion of lecture notes such as Introduction to Stable homotopy theory. I could imagine sketchy notes being much less helpful.
That the nLab is in large parts lacking good exposition and introduction is one thing. But that this is regarded not as a deficiency but as a virtue or even a must, I find perplexing.
Urs, surely you know nobody actually thinks that. Please, let’s not twist words or exaggerate. Nothing against introduction or exposition has been said anywhere as far as I know. In fact the exact opposite is true, at quite a few discussions we’ve had.
One thing I said earlier:
“For less advanced subjects like the page on the structure theorem for finitely generated abelian groups: examples are certainly okay, but here there are a million places to learn about that stuff, both online and in textbooks. I don’t see examples for relatively elementary topics like that as being so critical to include in the nLab.”
Clearly, and as I said, I am thinking of the audience, or what I think the audience is. This thread was started by my #5, where I express puzzlement over that example 3, and both Mike and David understood what I meant there. It was not an outrageous question. (And since it was understood, a suitable adjustment was made.)
(I’ll interpolate to admit that I, like David, never understood the point of the invariant factor decomposition, so on that particular presumably elementary point, I personally would benefit from some illuminating commentary. Maybe I’ll research that one.)
Let me give another example or perhaps thought experiment, which hopefully makes the point I was trying to make clearer. We have a page on the quadratic formula. I think maybe Toby started it. There’s actually some nice commentary there and I’m glad the article is there. But if an example were to be added, along the lines of “Here is how one would use the quadratic formula to solve the equation $17x^2 + 6x - 5 = 0$”, then I’d think that a little silly. One could try to defend that by invoking young people again, but seriously, for the nLab, I think that would be a little off-topic (or lacking in taste, or something).
How do you think young people are going to get to appreciate the sophisticated cutting edge stuff on the nLab if there is nothing that leads them there?
Of course there is not absolute disagreement, but if we’re talking about topics that arise in a first undergraduate course in abstract algebra, then of course there is quite a lot out there that prepares the ground, and it’s not necessarily the nLab’s job to be the pupil’s teacher in every respect.
There are many, many not particularly high-level topics I have touched on at the nLab. Like determinant, a topic that appears in a first course in linear algebra. My usual approach is to strive for something elegant and concise and from a functorial point of view, and something at more or less a graduate school level that is more or less a common core of mathematicians. Now if someone were to add a numerical example on how to use the determinant to check that a $3 \times 3$ matrix is invertible, then I think that would be taking it too far. The $\mathbb{Z}_{12}$ struck me as similar to that.
already then I didn’t understand why you, Todd, were being so sceptical.
I’m not sure how public I was about that, so maybe I’d like to keep this brief. At the time, to me it didn’t look like it was fitting in very organically with other parts of the Lab. I’d have to go back to see if I still think that. (But I still think such ventures should be labeled as lecture notes, if we want to include them.)
As to David’s bringing up mate, let me think about how to improve. There are lots of examples. For me personally, I am hampered by the fact that I am absolutely no good at graphics.
Hello all.
If I may, I like to present a outside perspective as someone who is not a contributor to the nLab but rather somebody who’s doing their best to understand the concepts presented therein.
As an amateur, I rely on your trivial examples.
I’m currently reading Mike’s “Catalog” paper and I am grateful for his attitude at the beginning. He specifically states that the trivial examples of unary type theory are seldom discussed… and yet those are exactly the types of examples that serve as a useful starting point or for somebody who’s outside the field to start to understand these concepts. So when Todd states in #13 that there are millions of places to find these “trivial” examples, here is a glaring example of that not being the case.
It seems to me that the crux of this conversation, as others have pointed out, has to do with the intended audience. One sector seems to think that the nLab is written for experts only and therefore trivial examples are of no use while the other side thinks that it is written for amateurs and thus experts don’t need as many examples.
I personally don’t see why it has to be one or the other.
As I think Urs said, there is no harm in including extra examples and much benefit.
In conclusion, as a complete amateur who is just now going to start reading the HoTT book, I implore that you keep putting up as many “trivial” examples as you can come up with!
“But if an example were to be added, along the lines of “Here is how one would use the quadratic formula to solve the equation 17x2+6x−5=0 17x^2 + 6x - 5 = 0”, then I’d think that a little silly. “
Yet this is the most common application of that formula, the one that most people will be familiar with, and thus the one that people are most likely to connect with.
It will not cost Nlab any extra money to include… no one is being forced to include that example… it doesn’t need to go through all the steps (just setup the equation, give the answer, and let the reader work out anything else for themselves)… it will not detract or confuse the entry… and you can tuck it away under a “trivial applications” heading at the end of the entry so as to not offend the more sophisticated reader.
I don’t find that silly at all…
One sector seems to think that the nLab is written for experts only and therefore trivial examples are of no use while the other side thinks that it is written for amateurs and thus experts don’t need as many examples.
Again, a misrepresentation. I do not think the nLab is written for experts only (that’s such a relative term, anyway). I do believe the nLab works best when written with some focus, i.e., with a view to exposing categorical POV’s (nPOV) for an audience with roughly graduate-school-level sophistication (but not the sophistication of the other type, who sniffs at category like a snob may sniff upon seeing an album of Chopin Nocturnes in someone’s collection; for the allusion, see here).
I mean, we could be giving examples of solving two linear equations in two unknowns, etc., or explaining decimal arithmetic to schoolchildren, etc. (I don’t mean carrying, which is written for a more sophisticated audience). Theoretically, there’s no limit. But going too far entails losing some focus in my opinion.
So when Todd states in #13 that there are millions of places to find these “trivial” examples, here is a glaring example of that not being the case.
Sigh. I wasn’t talking about that. I was talking about things like the structure of $\mathbb{Z}_{12}$.
I will say more later when I have more chance to think, but I just want to point out quickly that it is not true that “adding material never has a downside”. Namely, the more material you add, the less concise you end up, and conciseness is a virtue for readability. It’s not an absolute virtue — it’s one that has to be balanced with other virtues — but it’s not worthless. This is maybe similar to Todd’s point about focus.
If you have one group that writes nlab aimed at grad students, another at undergrads, and another at high school, then it should be obvious why focus will be hard to come by.
A grad student would have no use for an example of the quadratic formula, an undergrad may have use for it as a reminder, and a high school student may rely on it to learn it (in the context of the n-POV as a bonus).
Right! Trying to be everything to everyone leads instead to being nothing to no one. I generally see the nLab as at about the same level as MathOverflow: research-level mathematics. We certainly write about elementary subjects, but we assume the audience has the background of a research-level mathematician (including graduate students).
This is very different from the question of whether a page about any particular subject is written for “experts” or “newcomers” to that subject. I think we can and should assume a certain mathematical maturity, like the ability to plug in numbers to a formula oneself rather than needing to have it spelled out. That’s the sort of “trivial” example that, arguably, detracts from conciseness more than it adds to clarity for the intended audience. But examples that are “trivial” to experts in a field are, of course, still important for newcomers to that field, regardless of the mathematical maturity of those newcomers, and those are the sort of examples that we should of course include.
Since my categorical logic notes have been mentioned a couple times, let me say why I chose to write them in LaTeX instead of on the nLab. One reason is macros. For writing short pages on the nLab, I can deal with having to type \mathbb{N}
every time I want to refer to the natural numbers (or even to hit C-c C-f C-s N
in Emacs), but in a long work I would much rather be able to define a custom macro \N
. I also really value being able to define semantic macros, so that I can change my mind about how to notate something halfway through and not have to search-and-replace. Finally, the ability to draw diagrams with xypic and tikz, and derivation trees with mathpartir, has no equal on the nLab.
Another reason is complete editorial control. I am writing them from a particular point of view, which may be controversial in some circles, and I want the freedom to express that point of view without anyone else editing it, without it getting mixed with other points of view on other pages on the nLab, and also without implicitly attributing my opinions to anyone else.
There are probably other reasons, but those are the main ones that come to mind. That said, if anyone wanted to try importing some parts of them into the nLab once they are in a more finished state, I would certainly not mind.
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