Pedagogically, that will be clear but, I am `fond' of writing subtle proof such as this! Because, there are some cases in group theory, especially that in Symmetric groups, where such a straight forward proof is longer.
@Srivatsan The contradiction is not to a bigger fact that, we are considering a finite index subgroup, but to a otherwise relatively weaker part of the hypothesis that the finite index subgroup is also proper!
@Srivatsan I miss him so I thought about asking him to come over for lasagne but then I thought that's a very bad idea because if he's too busy to visit SE then being invited for dinner will probably put additional pressure on him.
@MattN I mean: close enough distance-wise. I can't imagine a dinner with you in the near future, because we are so far-apart. (Again: distance-wise :-/)
@MattN Over Christmas break my freshman year I stayed with a family that spoke Schwyzertüütsch at home. It was ... interesting. Dos isch min Fuass. And of course the Khukhekhästli.
@Skullpatrol No. That link pointed to a book on google books. I can't actually view the contents. But I have plenty of other things to do and wouldn't have time to read it anyway!
@Skullpatrol ???? Today I can read that page. I wonder if the difference is simply that yesterday I read a lot of previews, and today this is my first.
@Skullpatrol Nope: it was the same link before. I could even see the yellow block showing where the search term was, but the text itself didn’t show, and I got the unavailable/reached your limit message.
@BrianMScott The next couple of pages explain these so called "math wars" in education from the point of view of the author who worked in Stanford and is now at Sheffield in England.
@Skullpatrol I was just skimming them; I can wholeheartedly agree with what she says on the page with the heading Learning without Thought: I got a lot of those students in my courses at the university where I taught. I’m also delighted to see that she recognizes that different methods can be effective.
@BrianMScott I'm getting the same message as you got when you first tried viewing it: the yellow block showing where the search term (Learning without Thought) was, but the text itself didn’t show, and I got the unavailable/reached your limit message.
I thought that cosh sinh question at first was about memorizing the formula itself, so I thought of a silly mnemonic: $$\Large \color{Blue}{\mathrm{c}}\mathrm{osh}^2 \color{Red}- \color{Blue}{\mathrm{s}}\mathrm{inh}^2 \color{Red}= \color{Blue}1$$ $$\Large\rm \color{Blue}C \color{Red}- \color{Blue}S \color{Red}- \color{Blue}I$$
@Kannappan: Also, there is a shorter way to do the part where $H_1 \cap H_2$ is normal in $H_1$ and $H_2$. The normalizer has to be the entire group since it is a multiple of $2^5$ (lagrange) and of size larger than $2^5$ (contains two different 2-sylows)
@Kannappan: Actually this is my third year at the university.. it's kind of complicated. I take the courses I need/want to take. I had only three during the fall, but those were all pretty big ones (group theory, topology, analysis)
@Skullpatrol I understood the issue: it’s been around in one form or another at least since the 60s. I hadn’t been aware of the current flap in California, however.
I suspect that Jo herself is a good teacher, but I’m virtually certain that a lot of teachers would (with the best of intentions) make a royal hash of her approach.
@Skullpatrol No problem; I just meant that even without seeing the book, I had a pretty good guess at where it was going.
Sorry, Skullpatrol, but I think you are distorting Brian's words to suit yourself. That is clearly not what he meant. // OK: you are not really distorting, since you didn't quote him wrongly or anything. But I still don't see your point.
@Asaf: I don’t suppose that you have any idea how much choice is required for every infinite set admits a bijection with a set of the form $x\times\omega$, have you?
@AsafKaragila Are you using $\mathfrak{p}$ as a general infinite cardinal, or as the minimum cardinality of a subset of $[\omega]^\omega$ that has the strong finite intersection property but no infinite pseudo-intersection?
