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First, where can students find lists, information, or resources on the crucial results, inequalities, theorems, etc... which a textbook might not explictly feature or even bring up at all?

Second, because textbooks don't advertise this, how can students determine which textbook exercises are really fundamental to the subject?

Here's some context. While discussing subgroups with an undergraduate, I mentioned the One-Step Subgroup Test. He said he didn't know what it was. I was confident the course textbook (the one by John B. Fraleigh) had to cover it. I looked at the chapter on Subgroups but didn't see it. I had to pore over the textbook exercises to see it shrouded as exercise 46 on page 58 in chapter 6:

Show that a nonempty subset $H$ is a subgroup of a group $G$ $\iff ab^{-1} \in H$ for all $a,b \in H$.

I showed this to him the next time and he said he remembered doing this exercise. However, he didn't realize what it really represented. I don't think he's at fault. I'm very disappointed that the textbook didn't pitch this information or at least a sentence or two on the intuition or motivation. Why don't textbooks trumpet this? Therefore, how students can efficiently find out about essential results that textbooks overlook, shrug off, or displace into an exercise with little or no motivation?

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  • $\begingroup$ So, yes, a texbook should state this as a theorem in the main text and write that the proof is an exercise. Also yes, textbook authors (just like students and other people) do not always do what they should. What kind of answer are you expecting except "you are right"? Try to use good textbooks. Like your undergraduate, talk to other people about mathematics. Accept that there is no way to "efficiently" learn all mathematics. (There are results that are cited in research articles as (book, author, solution to exercice 3.125). $\endgroup$ – Phira Dec 31 '13 at 16:45
  • $\begingroup$ It frustrated me a lot during my last high school years and my first university years that it was next to impossible to find out which books from the mathematics library I could read without actually reading them. But there is a reason for this state of affairs. $\endgroup$ – Phira Dec 31 '13 at 16:48
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    $\begingroup$ I have changed country and now I am teaching to first year university students "extremely fundamental theorems" in linear algebra that I have never seen before and never missed. On the other hand, some "extremely fundamental theorems" in linear algebra that I learned as a student are missing from the program. How should a textbook author do better? $\endgroup$ – Phira Dec 31 '13 at 16:49
  • $\begingroup$ @Phira Thanks for your comments. I'm after answers that can help with this problem. I'm certainly not after whether anyone's right. $\endgroup$ – Pamela Lee Jan 22 '14 at 5:27
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One way to detect that an exercise has some importance is if its result is used later in the text.

As for lists of crucial but omitted results... well, I doubt such lists exist, and if they did, they'd omit things too, and people would object to those omissions. Not because every subject has a canonical list of crucial results and authors neglect to consult it, but because what's crucial and what's not is often a matter of opinion.

(To take your example: in my opinion, the one-step subgroup test is a minor technical trick which occasionally saves a bit of time but has no conceptual significance whatsoever; as such, I don't think it deserves much attention, and am not at all shocked by its treatment in the text you describe.)

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  • $\begingroup$ Excellent answer! $\endgroup$ – Pamela Lee Mar 9 '14 at 7:33
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I'm not sure there is an efficient way of finding out. The best way I know of is to consult a variety of textbooks. Different authors have different ideas of what is important.

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