Hm, you are invoking a mini-contradiction in the end. I am wondering whether one could avoid it.

Maybe instead of starting off with the assumption that H is a proper finite-index subgroup and arriving at a contradiction,

you could start by assuming that H is a finite-index subgroup and prove that it is R.

And hope that it is obvious at this stage that no such proper subgroup exists.

This is slightly iffy; I understand if you disagree with my suggestion.

(And, a minor typsetting thing: the "$R$" in the fourth line should be an $\mathbb{R}$.)

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.

@KannappanSampath Why do you think one of these two alternatives is more subtle than the other?

hi Matt.

Hi Srivatsan.

I was mentioning you yesterday(?); see here: chat.stackexchange.com/transcript/36?m=3160327#3160327.

@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 The exam is on February the 14th or 15th, can't remember which exactly.

@KannappanSampath Hm, you just rearranged the hypothesis; I don't see the difference.

@MattN Ah, I thought you had a fun-analysis exam in end January.

I hope it is as much fun as the name suggests.

@Srivatsan Which I cancelled. I'm taking that in summer.

@MattN Oh.

So which one is this Feb. exam?

Logic and Set Theory.

(I cancelled some other exams too)

In fact all but this one.

: )

You are doing fine, right?

May I know what is EIR exam?

@KannappanSampath - I think you are a high school student from i know you read the 101 trig problem

@Srivatsan With exam preparation for set theory?

Well, if I were to guess: his name sounds like he is a first-year college student in Bangalore.

@Victor Haha, I was a high school student when I started thinking about that problem after reading the book! Now, I am a freshman, in US terms!

@Srivatsan LOL!

My name suggests I am a first-year college student. Then next year, if it suggested the same thing?

@MattN In general. :=) But wish you good luck with the coming exam.

@KannappanSampath Oh, perhaps I check the stars before making such a prediction. Surely they change positions, don't they?

@Srivatsan You are thoroughly logical, @Srivatsan.

Wow, sometimes it's hard for me to grasp that you are still a freshman.

You're amazing!

I know I am.

Also, @Srivatsan could acknowledge

@Srivatsan I guess...

@AsafKaragila Everyone knows that, Asaf. That I am amazing is more surprising, though. :=)

@AsafKaragila Is this like marking your territory?

@Srivatsan Well, you have so many deities... it's amazing :-P

@MattN Nah. It's like burping.

@Srivatsan And thanks. : )

@AsafKaragila Ah, you seem to think of my oh-so-many-deities more frequently than me. =)

@Srivatsan Yes, I thought to convert to Hinduism but then I realized that I only have a countable number of arms... :-)

Yes, we have a strict lower bound on the number of arms. You wouldn't have passed that test.

@Srivatsan Did you see that we've had a fluffy visitor today?

@MattN Who? :)

Nine-fingers?

@Srivatsan Well how many fluffy ones are there? I can only think of one.

(I didn't mean in here)

Is it ok for me to look clueless now?

No : D

Btw: why is Ninefingers fluffy?

Perhaps because he’s a hobbit?

Oh I thought he said he wasn't.

My bad. Ok, so there are two candidates in total, one remaining.

@MattN I wouldn’t know either way, but the name is suggestive.

@Srivatsan Still clueless?

Yes. :/

I have earned a Nice Answer badge. Thank You @Srivatsan and @ymar for making substantial improvements in the answer!

I’ve not been following closely; is that the 160 question?

@KannappanSampath You're welcome. :=) (But I had nothing to do with your Nice Answer badge; I swear..)

@Srivatsan Here's a hint: it starts with teddy and it ends with bear.

@MattN What?! Fluffy was here?

@BrianMScott 160 question is dead now! No one probably liked the fix Except Dylan!

@Srivatsan Yes!

@MattN Did he come to chat?

I was thinking about him just now actually. I was wishing he came back at least for a little while this week.

@Srivatsan I don't really know what he did but his profile says he was here.

Maybe to write "on leave" in his profile :=)

@Brian It is this

I take that Fluffy is Theo?

@BrianMScott Ya, apparently. [I am just playing along with Matt. =)]

@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.

@Srivatsan Actually you invited the name! : D

@MattN Oh, are you guys close enough to meet for dinner?

@Srivatsan How close does one have to be to have dinner together?

@KannappanSampath Very nice.

@MattN Yes, but I didn't know bears were fluffy. Oh I see now, this is a teddy bear.

@BrianMScott The other day someone pointed out what the initials t and b stand for.

@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 :-/)

@Srivatsan Nice save!

@Srivatsan I misread you. blushes

@BrianMScott Save? That is exactly what I meant... :=)

I should've said: "Do you guys live close enough..."

@MattN If there is a drive of at most 15 minutes involved then it's close enough for dinner.

To answer the question: I think we are in walking distance of each other. (It's safe to say this as this place here is fairly small)

Otherwise it's a special occasion.

@AsafKaragila I read relationship-wise.

Huh?

Well, that's not what I meant. It's a dinner, after all.

@MattN This place being?

@AsafKaragila When Srivatsan asked me that question.

@Srivatsan ?

@MattN So how are you with the local dialect?

@MattN Just ignore it. I just repeated myself..

@all: A question is receiving too many downvotes. Looks like it could be closed on some grounds!

this is that question in question!

@BrianMScott Peasant? I'm fluent. Why? : )

@KannappanSampath I just supplied the fifth vote to close, so it’s dead.

@BrianMScott Thank You, Brian!

@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.

@BrianMScott That's a surprise! Not bad! So you speak it?

seems to be close to be closed need two more votes, who wants to help?

@MattN No, and nowadays I wouldn’t understand it if I heard it, but I can usually work it out if I read it.

@BrianMScott Of course, the obligatory "Chuchichäschtli". : )

@Ilya Executioner needed!

@Srivatsan Done.

$5$ more votes and I'd cap for the day! hohoohoooo....

@MattN Ah, yes, I’d forgotten the sch. (But I’ve a good excuse: that was Christmas of 1965.)

I am casting reopen vote here: math.stackexchange.com/questions/102739/…. If others want, they could too. 2 more votes needed.

The question has already been answered, against all odds.

There wasn't a single hint that it had anything to do with combinatory logic!

@BrianMScott You're excused. And anyway: as it's not a written language you may spell it as you like.

Maybe people who have read the book might know "Ky". That seems to be the only piece of hint in the question.

@MattN I meant that I’d forgotten the pronunciation.

@Srivatsan I’d have got it from the revised version; it’s a neat book. I just cast the final vote to reopen.

We talked about it in this chat a few weeks ago, in fact.

I see. In fact, that it is written by Smullyan suggests so already. =)

That's the only reason I know about it.

@Srivatsan Second that.

@Kannappan: That question was rhetorical =)

And, SE goes offline! @Srivatsan

right, I just noticed.

Wanted to write the answer to : Given $2^xx^2=1$ solve for $x$.

@Dylan Welcome back!

So, some counting killed 160!

It did.

Sorry for not being of much help. I kept getting distracted.

@DylanMoreland That is no problem. But I was sad that that answer that took long me long to think never got upvoted!

Well, upvotes aren't worth very much :)

@MattN Matt did you get a chance to form an opinion on the "math education" wars link I put up?

You helped someone and thought hard about something; that's pretty good already.

@DylanMoreland Agreed!

I was going to upvote it once I had the chance to read it fully. It seemed like it was on a good track though.

@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!

@MattN np

@KannappanSampath Annoying: I discovered it by posting an answer!

@BrianMScott Holy ...., You'll have to retype the whole answer!

Yikes.

@MattN I think that Skullpatrol meant this link.

@KannappanSampath And it was fairly long, though elementary.

@BrianMScott Uh oh

@BrianMScott Oops!

@BrianMScott Back button didn't work out?

Hmpf. Upgrading curl breaks awesome.

Why wasn't there an alert?

@BrianMScott First impression: design is off-putting. Second impression: the word "scientists" in the first sentence I read is even more off-putting.

@AsafKaragila do you spawn curl anywhere in your .awesomerc?

@DylanMoreland No, it’s okay; I just hadn’t checked yet when I wrote the previous comment.

@BrianMScott Thank you Brian I got that link out of the book on google books

@BrianMScott It might have been saved.

@Daniil Hmmm, good question. I'll check.

@MattN Even if it isn’t, it’s currently sitting in my browser.

Yeah, for the MPD widget.

But none of the previous upgrades broke awesome, nor a reboot fixes the situation.

Amazingly enough, this is only the second or third time that I’ve been online when SE went off.

@AsafKaragila well try disabling that widget

Huh. It doesn't use curl.

Perhaps vicious or something uses curl.

@BrianMScott This is the link that I'm wondering why nobody can gain access to? books.google.ca/…

Yeah, vicious uses curl for network stuff.

Let's see if there is a git update which would solve that.

@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.

@BrianMScott Great!

@BrianMScott Similar experience here. But I don't read too many previews from g. books, so that is probably not the explanation?

@Srivatsan Dunno. shrugs

shrugs back

Maybe I made a mistake some where :-(

in copy & paste.

@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.

@Daniil The git log of vicious does not indicate any changes in the past month that would correct this behavior.

@AsafKaragila but have you narrowed the problem down to the widget?

Well, this is the only package I have installed which depends on curl and related to awesome.

well maybe you can hack it to work with the latest curl?

frankly if you have installed curl and that widget from your distro's repository you should file a bug report

Well, the problem is that curl is from the repo but the awesome stuff is from AUR so there's no official support there.

@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 Is that Learning without Talking?

No, it’s as I wrote it; it’s three pages after the one to which you linked.

@AsafKaragila ah, archlinux... brought to you by the folks who updated the python to version 3 breaking all the backwards compatibility :P

@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.

@Skullpatrol Sounds like some sort of conservation principle at work. :-)

@BrianMScott Yes, the page right before it explains the aim of the book.

@Brian: I wrote an answer to a notation question. I need your help in figuring out if its concise and to the point! Here is it

Kannappan, is that symbol actually used somewhere?

never seen that before

It's used in recent papers in the literature. MO has an interesting question on Notation, in which someone introduced this! @mk.

@mk Did you see my Fix for 160?

I thought that cosh sinh question at first was about memorizing the formula itself, so I thought of a silly mnemonic: $$\Large \color{Blue}{\mathrm{c}}\mathrm{osh}^2 \color{Red}- \color{Blue}{\mathrm{s}}\mathrm{inh}^2 \color{Red}= \color{Blue}1$$ $$\Large\rm \color{Blue}C \color{Red}- \color{Blue}S \color{Red}- \color{Blue}I$$

@anon May be people might like it! Post it you'll get my +1!

It's technically not part of the question, though.

@Kannappan I saw it, but I'm not sure about that last part yet

Anyway @Kannappan, do you know what "CSI" stands for?

@anon Crime Scene Investigation!

@mk do you still think there is something wrong?

Speaking of which....

Also, I would mention why we can find $H_1$ and $H_2$ s.t. $|H_1 \cap H_2| > 1$

@Kannappan: I don't know, I haven't thought about it yet

@KannappanSampath Like m. k., I’d never seen that symbol before.

Whoah, that is a sweet symbol for "acts on." Is that real or did you think of that?

This suggests it! @anon @BrianMScott @mk

They’ve changed the context menu when you right-click on a MathJax expression.

To view source, you need to click on View Math as>TeX Codes!

@Daniil Yeah, but other than that I find the system excellent. Plus there were no complaints about curl, so I'm probably a special case.

@Brian Oh, I thought it was just some bug on my browser

more effort to see the source..

@anon @Brian @mk This gives a notation for characteristic subgroups!

@mk That was my reaction, too. But perhaps the extra options will help some folks.

@KannappanSampath Not a bad one, though I doubt that anyone to speak of is using it.

But, it will become standard in a few years as it travels along!

@Brian

@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)

@mk But why multiple of $2^5$?

since it contains $H_1$, and $H_1$ is a subgroup of order $2^5$..

also, that second link is the same one

my teacher used H char G to denote that H is a characteristic subgroup of G, I think that one is pretty common

It is, but isn't this cuter, $ \triangleleft !$

I agree on that one about the first case!

@mk I’d go so far as to call it standard; it’s even used here.

@mk

I might be remembering this wrong, but I think I saw somewhere $\blacktriangleright$ for characteristic subgroup

$\blacktriangleleft$

But, practically you need to shade, so I'd think $\lhd!$ is better

@mk

@Kannappan you're right about that, it isn't practical

@mk BTW,If I may ask, where do you study?

I study at a university in Finland

@mk You are a graduate student, aren't you?

no, I'm still undergrad

@mk Oh, I see!

i switched from CS to maths last fall

@mk So, you can do that anytime?

not really

you can apply twice per year

@mk So what are the courses you have had?

there are many

@Srivatsan Are you enjoying the google book link Srivatsan?

@Skullpatrol not really. I don't agree with half the things the author says.

but some algebra, group theory, analysis. I'm still missing a lot of basic stuff. Studying probability right now.

@mk Oh, given you are still in your first year, you must have had 5 courses a semester, no?

@Srivatsan What do you disagree with most?

@Skullpatrol I don't want to get into this. I guess I am not into this education thing.

@Srivatsan np

@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)

oh fine, So, I may not understand your system! Any way, you do group theory well!

Well thanks. I do like finite group theory a lot

@BrianMScott Does reading the google preview illustrate this issue, as I was trying to do last time?

I wanted to take a look at that book skullpatrol linked, but google books refuses to let me preview it

@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.

@mk Google books refuses to let a lot of people preview it ... I don't know why.

@BrianMScott I didn't mean to imply that you don't understand it, sir.

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.

@BrianMScott May I ask what a "royal hash" is?

@Brian: Should we delete it? Cast a vote? This question: math.stackexchange.com/questions/103343/…

@Skullpatrol A real mess.

@Srivatsan: that says "please delete"

@mk Doesn't mean we should listen to the OP :)

Royal hash: When you use a mathematical one-way shuffle of face cards in a deck.

hm

$1$ vote to cap! Exciting!!!

I don't see the other 20k+ voters around. If Asaf were here.

@Srivatsan Does it really make no sense?

@BrianMScott In my opinion, which doesn't mean much, mathematics gets royally hashed as soon as it is handed down from the mathematicians.

@BrianMScott I am not sure, I am still reading it now.

Actually, I think the question is fine. In which case, it should be rollbacked.

@BrianMScott Qiaochu deleted it just now.

Saves us the bother of thinking!

@BrianMScott "Learning without Thinking" ... sounds familiar ;-)

I never think.

@JonasTeuwen Me neither. That's why I decided to do maths! To learn it. : )

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.

@MattN Exactly.

@Srivatsan Yes, but it was a funny distortion.

The only time this room ever makes any sense is when we're talking about math. And even then.

@Srivatsan It was a joke!

Learn so you don't have to think

@Skullpatrol Thanks for the clarification. (I guess I needed it since I obviously didn't take it to be one. :-))

@Skullpatrol And very appropriate, given the context.

@BrianMScott Thank you.

@BrianMScott Hm, to think I was "defending" you... =)

I guess I will be off now. I have to get lunch anyway.

@Srivatsan See you later!

@Srivatsan You would never have to defend anyone against me, because I'm perfectly harmless ;-)

Oh, sorry. I didn't mean it that way.

np

= no problem

Right. Food is cooking.

I'm bathing chicken.

I hope that it’s dead; might be a bit too exciting, otherwise.

: )

Oh yeah, I'm not sure a live chicken would be happily marinated in olive oil, dates syrup, soy sauce, some mustard and a bunch of spices.

@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?

@BrianMScott Hang on.

Well, DC+$\frak p=2p$ should be enough, I think.

@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?

General infinite.

@BrianMScott Isn't that $\frak b$ or $\frak d$?

What's DC?

No, $\mathfrak{b}$ is the minimum size of an unbounded subset of $^\omega\omega$, and $\mathfrak{d}$ is the minimum size of a dominating family.

Dependent Choice.

@MattN Dependent choice.

Thanks!

@BrianMScott Ah. I didn't know there was another fraktur font cardinal. ;-)

I know of six: these three, $\mathfrak{c}, \mathfrak{s}$, and $\mathfrak{t}$.

Oh yeah $\frak c$ :-)

Oh, and there’s also $\mathfrak{a}$.

Either way, if we have DC and $2\frak p=p$ then $A = (A+A) = (A+A+A) = \ldots= A\times\omega$.

Minimum sizes of a splitting family, a tower, and a MAD family.

@AsafKaragila Looks good to me. I’ll add that to my answer to the permutation question.

I'm not sure whether or no we can crunch it to less than DC (ACC or something?)

Hang on, though.

I'll see if I can get a nice bound on both of them.

Or perhaps I should write a $\mathfrak{s}$plitting family, a $\mathfrak{t}$ower, and $\mathfrak{a}$ MAD family.

Well, there's no nice bound on those.

I'm gonna finish with the chicken.

Chicken! :-)

I’m going to finish that edit and then do some non-mathematical chores. sigh

@BrianMScott Could you give me an example of a mathematical chore please?

Working with category theory. ;-)

Justifying the steps in an argument.

Thank you both.

$1$ upvote; $3$ hours!! Nail-biting!!!