Mathematics - 2015-06-06 (page 2 of 5)

Paul Plummer

07:00

I probably could not S++ every level in that game, even if I put 1000hours in, but I think I could A+ every level in SMB in less than 1000hours

Mike Miller

Here's an old version of the best times on everything

I have SS'd every level but one... it burns

Paul Plummer

Oh seriously!

Remember me

Hello@Balarka!!!

Mike Miller

Well, all the normal levels. It then opens a room with "Megadifficult", "Gigadifficult", etc

I think Giga is the one towards the achievement so that's the 'one' I haven't SS'd

Only one of the crazy ones I've beaten...

Remember me

Well I dont want to ask this question on main so I am asking it over here
Given two metric spaces X and Y and given that f and g are continuous mappings from X to Y and f(x)=g(x) for every x in A where A is a subset of X. Prove that f(x)=g(x) holds true for every x in $A$closure

@PaulP can i get some help from you?

Is it fine if I just prove f(x)=g(x) for some $x\in A'$ where $A'$ is the set of limit points of $A$

Balarka Sen

07:06

Hmm, it looks like there might be more about the directional derivatives being linear for differentiable function than meets the eye. I guess we can do something wacky forgetting about what $D_{\vec{v}}\vec{f}$ says about $f$, and concentrate on the $\vec{v}$?

Remember me

Hello@Balarka Are you free?

iwriteonbananas

@MikeMiller if $f$ is an entire function that is not a polynomial, why does the functoin $g(z)=f(1/z)$ have an essential singularity at $0$?

Balarka Sen

I am not, @Remember

Remember me

Okay fine

Balarka Sen

@Remember hint : draw a picture. don't forget what limit points and continuous functions mean.

Mike Miller

07:08

@iwriteonbananas Remember the classification of singularities. Show that the assumption that f is not a polynomial implies |f(z)| does not just blow up as you go to $\infty$.

Since it's also clearly not a finite value at infinity it must not be a removable singularity.

iwriteonbananas

@MikeMiller did you mean $|g(z)|$ does not blow up for $z\to \infty$?

Mike Miller

I meant |f|. Equivalently I meant |g(z)| does not blow up as z -> 0.

iwriteonbananas

right, my bad

well, that's what i've been trying to see, but i havent been able to.

Remember me

07:58

Ahh I still cant get it

Balarka Sen

@Remember $\{a_n\}$ be a sequence converging to $a$. If $f(a_i) = g(a_i)$ for all $i$, is it ever possible that $f(a) \neq g(a)$, where $f, g$ are continuous functions?

There should be nothing so hard about the exercise once you know why it should be true.

Remember me

@BalarkaCant you do it by using definition of continuity and idea of limit points?

Balarka Sen

so why in the world aren't you doing it?

Remember me

I am trying my best to do it ... I dont like sequences and i dont want it to be done through it..@Balarka

Balarka Sen

well, probably you don't like sequences because you haven't studied it well enough.

aren't you doing Rudin?

Remember me

08:09

That might be a reason...
But using open balls and stuff its better at questions in continuity I feel

No simmons@Balarka

Balarka Sen

ok, if you want to do it by open balls definition, you can do that too.

all you need is the definition of continuous functions, really.

Remember me

that
$f(B_\delta(x)) \subseteq B_\epsilon (f(x))$?

Balarka Sen

without appropriate quantifiers, that statement makes no sense.

Remember me

Okay okay

Balarka Sen

but yes, you can use the pullback of open sets is open definition to do it.

Remember me

08:12

Ahh thats one theorem which i proved myself

Balarka Sen

which theorem?

Soham Chowdhury

Balarka, where did you study analysis from?

Balarka Sen

I didn't study a lot of analysis, @Soham.

Soham Chowdhury

The little you have.

Balarka Sen

did a first few chapters of Rudin once, and that's about it.

Soham Chowdhury

08:13

Oh.

Remember me

If F is continuous then f^-1(G) is open whenever G is open @Balarka

Well with the metric spaces

and all

Balarka Sen

@Soham When are you going to meet Samik?

Soham Chowdhury

Kichhu bole ni specific.

Remember me

Whose Samik@Soham

Soham Chowdhury

He said he'll inform me when he gets back.

@Rememberme A prof

Balarka Sen

08:17

ah, okay.

Remember me

How do you guys get to meet prof... !!

I have never met one

Soham Chowdhury

@Rememberme I . . . worked it out. ;)

Remember me

How

Soham Chowdhury

Gugel shows the way.

Balarka Sen

glares at @Soham

Remember me

08:18

Wha??

Soham Chowdhury

I have a black belt in Google-fu.

Balarka Sen

we don't want the uni to be filled with 15 y.o. grad students, so shaddup.

Soham Chowdhury

"we"

Grad students?!

Remember me

Why?? and Grad students !!

Soham Chowdhury

And correction : I'm 16.

:)

Balarka Sen

08:19

You're in 11th grade?

Soham Chowdhury

Yeah.

:(

Doing a lot of mult calc?

Remember me

Ahh Soham cmon tell me how do you get to meet these people

Soham Chowdhury

Where do you live?

Remember me

Bangalore

Balarka Sen

@SohamChowdhury yeah, kind of.

Soham Chowdhury

08:21

And I haven't met anyone yet.

Balarka Sen

@Remember lot of good math guys in IISc. mail one of them, I guess.

Remember me

You will

Soham Chowdhury

@Rememberme Find the websites of the unis close to you and email the people there. Exactly what I did.

Remember me

Ahh .....

Soham Chowdhury

@BalarkaSen CMI is not in Bangalore

Balarka Sen

08:22

I never plucked up the courage to mail anyone :P

Remember me

Balarka you are the master.... people call to talk to you :p

Soham Chowdhury

@BalarkaSen Spectral sequences guy/gal?

Balarka Sen

@Soham oh, I mean, I've never mailed anyone to get them to talk to me :)

iwriteonbananas

@DanielFischer if $g$ is an entire function, why does an estimate $|g(z)| \leq | \sqrt{z} |$ force $g$ to be constant?

Balarka Sen

My schoolteachers did it for me, I think.

Soham Chowdhury

08:26

@BalarkaSen You have nice teachers is all I'll say. :)

Balarka Sen

sure thing.

Soham Chowdhury

Anyway. Bunch of slick proofs in the cosets/quotients section.

Balarka Sen

oh, speaking of it, T. Chow mailed back. finally.

Remember me

@BalarkaSen Ahh Now I get what you mean

Soham Chowdhury

Tim Chow?!

Balarka Sen

08:27

yeah, author of the article on spectral sequences I was reading.

Soham Chowdhury

well.

Soham Chowdhury

08:40

why the "hence . . . " bit?

@iwriteonbananas there?

@Balarka?

Daniel Fischer

09:03

@iwriteonbananas If a holomorphic function grows too slowly near an isolated singularity, then the singularity is removable. Here the isolated singularity is $\infty$. You can see that for an entire function by the Cauchy estimates, for $\lvert z\rvert \leqslant R$, we have $$\lvert g'(z)\rvert = \frac{1}{2\pi} \left\lvert\int_{\lvert\zeta\rvert = 2R} \frac{f(\zeta)}{(\zeta-z)^2}\,d\zeta \right\rvert \leqslant \frac{1}{2\pi}\cdot 2\pi(2R)\frac{\sqrt{2R}}{R^2} = \frac{2\sqrt{2}}{\sqrt{R}},$$

and letting $R\to\infty$, you see $g'(z) = 0$ for all $z$.

Soham Chowdhury

Uh, @DanielFischer, can you help at all?

Daniel Fischer

@SohamChowdhury What with?

Soham Chowdhury

why does aH = Hb for some b imply aH = Ha as well?

as cosets?

Daniel Fischer

@SohamChowdhury Since $a \in aH$, from $aH = Hb$ it follows that $a\in Hb$. Also, $a\in Ha$, and the cosets $Hb$ and $Ha$ are either disjoint or identical. Since the are not disjoint, we have $Hb = Ha$.

Soham Chowdhury

ah.

thanks.

Daniel Fischer

09:08

You're welcome.

JC574

Hey guys

basic question about probability someone asking me -

if A, B are events

is B|A always independent of A?

I think so...

but I'm not certain it makes sense, haven't done prob in a while

Alec Teal

09:34

@DanielFischer @SohamChowdhury maths.kisogo.com/index.php?title=Coset for future reference.

iwriteonbananas

09:49

@DanielFischer good point, thanks

@BalarkaSen when you told me to construct a surjective self map on $S^n$ of degree 0, is this what you had in mind? $S^n \to D^n \to D^n/ \partial D^n \cong S^n$ where the 1st map projects everything down (i.e. "forget the last component") and the 2nd map is the quotient map

The Substitute

off topic - what does a weighted sum of 1 have to do with convex hulls? I understand the definition of a convex hull a set of points as the smallest convex set containing the points, but I don't see where a weighted sum of 1 enters the discussion.

Balarka Sen

10:16

@iwriteonbananas What d'you mean by "projects everything down"?

You mean projecting both of the upper and lower hemispheres onto the disk bounded by the equatorial circle?

But then your map is of degree $2$, not $0$, right?

Soham Chowdhury

Ah, cool. I finally understand why $\Bbb Z/n\Bbb Z$ is the One True Notation.

Balarka Sen

That's exactly how I felt after I learnt quotient groups.

Soham Chowdhury

somewhat unsurely raises hand for a high-five

Quotients and subgroups are done (somewhat)! :)

Balarka Sen

shakes head hopelessly kids these days.

2

@Soham Prove that $Q_8/\Bbb Z_2 \cong V_4$.

Soham Chowdhury

you don't really know something until you do the exercises - Balarka

Almost inb4.

Balarka Sen

10:27

oh, btw, you didn't compute $\mathsf{Aut}(D_4)$.

Soham Chowdhury

@BalarkaSen Trying.

Balarka Sen

after that, you had to do the harder $\mathsf{Aut}(Q_8)$

Soham Chowdhury

Yeah.

Balarka Sen

(fun fact : the latter can be done geometrically)

Soham Chowdhury

$Q_8/\Bbb Z_2$ is hand-doable. :P

(I think)

Balarka Sen

10:30

well, sure.

do you know the first isomorphism theorem yet?

Soham Chowdhury

no.

next section.

Balarka Sen

ok, prove that one yourself.

Soham Chowdhury

what?

Balarka Sen

$f : G \to H$ be a homom. prove that $G/\ker f \cong \text{im} f$

It's not that hard. If you understand quotients, you can do that without much voodoo

of course, you have to show that kernel is a normal subgroup and all

Soham Chowdhury

done that.

$\varphi(gng^{-1})$ etc

Balarka Sen

10:33

good

Soham Chowdhury

{1,-1} under multiplication is iso to $\Bbb Z_2$ na?

Balarka Sen

yeah

do the first isomorphism theorem, then that Q_8/Z_2 \cong V_4 thing follows as a corollary

iwriteonbananas

@BalarkaSen yes, that's what i mean. why would the composition have degree 2? it factors over 0 since $D^n$ has homology 0 for $n\geq 1$

Balarka Sen

it doesn't factor over anything at the homology level. how do you define degree of maps between non-sphere things?

as for why it's 2, note that fiber over each point has cardinality 2.

iwriteonbananas

@BalarkaSen homology is a functor, so it preserves composition. the composition is a map $S^n\to S^n$ but it factors over $D^n$, which in homology is 0

Balarka Sen

10:43

oh, nevermind my cardinality of fiber argument. i was lying.

hmm. ok, I agree that your map has degree 0

iwriteonbananas

ok

Balarka Sen

@iwriteonbananas :P i was confused by your composition of things. actually, the map I had in mind is the same as yours, but I'd have put it in a different way

iwriteonbananas

how would you put it?

Balarka Sen

namely, take S^n, map it to S^n by sending the upper hemisphere to all of S^n by pinching the equator and then the lower hemisphere to all of S^n similarly by pinching the equator, but compose with the antipodal map before this time.

this has degree 0 because it's nullhomotopic

another equivalent way too put this would be to take a nullhomotopic surjective map $S^1 \to S^1$ (given by the word $aa^{-1}$, say) and suspend $n$ times.

suspension preserves degree (why?)

iwriteonbananas

what does the map given by the word $aa^{-1}$ do?

hmm nvm

exactly what u described above, right?

Balarka Sen

10:48

yes, precisely

iwriteonbananas

i havent learned about suspensions yet, so i cant answer ur question

Balarka Sen

it's in Hatcher

Soham Chowdhury

well, if A = {1, -1}, $Q_8/\Bbb Z_2 = \{A, iA, jA, kA\}$, and that last set is iso to $V_4$.

works?

iwriteonbananas

why is your way of defining the map the same as the composition i described above?

Balarka Sen

@SohamChowdhury sure

Soham Chowdhury

10:50

ah.

i feel like I "get" why it does. :)

Balarka Sen

@iwriteonbananas look at what you map does restricted to the upper hemisphere

and then look at what it does to the lower one

iwriteonbananas

ohh yeah

Balarka Sen

it's precisely what I am doing with my map

Joshua A

What are you working on?

iwriteonbananas

true

that's a cool alternative way of seeing it

Soham Chowdhury

10:52

@BalarkaSen I want to see a bunch of simple (not the technical way) quotients of groups I know. any idea where I can find a few examples?

Balarka Sen

yeah, discovery of that got me a compliment from Ted saying my geometric intuition is better than him (although I am still not sure if it was meant as a sarcasm), @iwriteonbananas

@SohamChowdhury Dummit-Foote

loads of examples there

Soham Chowdhury

hmm.

iwriteonbananas

hahha

Soham Chowdhury

let me dig out my not-so-legal copy.

Balarka Sen

you're slowly turning into jasper, as already has been said

Soham Chowdhury

10:54

are those examples from things I'll know?

Balarka Sen

yep

Soham Chowdhury

oh, well, I'd like to think I'm helping you out in some way.

really.

Balarka Sen

i was just joking

Soham Chowdhury

and it hurts my eyes.

there's that, too. :P

okay. I've opened DF.

well. this is a big book.

in fact, I think I'm going to look over this book to kind of ensure there's nothing incomplete about what I've learnt. here goes.

iwriteonbananas

11:51

it's hard to focus, it's so friggen hot here

Soham Chowdhury

lol

iwriteonbananas

i'ts the same tempereature here

are you from kolkata?

Soham Chowdhury

yes, I am.

oh, wait.

are you too?

iwriteonbananas

hell no

Soham Chowdhury

well, that reduces the number of places by one. :P

iwriteonbananas

11:58

indeed indeed....

so you're from the same place as balarka?

Soham Chowdhury

aye.

i was actually guessing that you know @Balarka irl before your "hell no".

iwriteonbananas

kolkata is far far away from me

Soham Chowdhury

well, that puts a "possibility metric" on the set of places

iwriteonbananas

meh

Soham Chowdhury

not a metric actually. but still.

iwriteonbananas

12:01

what are u studying atm?

Soham Chowdhury

what are you doing? complex?

iwriteonbananas

no, im procrastinating

because i need to compute this integral

Soham Chowdhury

I'm reviewing the group theory I learned (till quotients) from Dummit-Foote.

put the \s in

iwriteonbananas

no, i dont care

i'll just delete it

Soham Chowdhury

bananas pls :'(

iwriteonbananas

12:03

it's too ugly

Soham Chowdhury

what do you need this for?

iwriteonbananas

analysis on manifolds course

Soham Chowdhury

oh.

iwriteonbananas

divergence theorem etc.

Soham Chowdhury

mm.

@Balarka look at this. $Q_8$ is not the only one.

skill patrol

12:11

hi guys :D

et al

Mary Star

Hello @skillpatrol !!

Could someone of you tell me why it stands that $R_x+F_{W_0, x}=0$ and $-R_y+F_{W_0, y}=0$ at the question physics.stackexchange.com/questions/187539/… ?

skill patrol

@MaryStar Hi pal, how are you?

Mary Star

I'm fine... I am preparing myself for Monday...

How are you? @skillpatrol

skill patrol

@MaryStar Fine thanks :-)

Mary Star

:-)

Remember me

12:39

Hello@Soham

Soham Chowdhury

hi.

what's up? still on your break?

Remember me

No.. no Finished continuity today

with exercises

Think of this @Soham:
Can a mapping be closed or open?

Soham Chowdhury

a function can be closed or open, if that's what you mean

Remember me

Yup mapping~function

user147690

Sad doing past assignments for revision and solving the whole things in 30 min, when you got like 75% the first time around...

Remember me

12:42

Hello@AlexC

user147690

Hey Sayan

Remember me

Its better we are studying on our own no pressure of assignments at least @Alexc :p

user147690

@Rememberme Indeed! And I suppose that it is good that I have learned the content then :)

Remember me

Yes...

So @Soham So what do you think can I say a mapping from two metric spaces X and Y which is continuous open or closed??

Soham Chowdhury

@AlexClark I know that feeling :(

@Rememberme Yes, I think I saw that in Bredon. Let me finish revising from DF and then I'll hit that book again.

Can't remember.

Abdou Abdou

12:47

hey everybody i want you to judge me this math.stackexchange.com/a/1308281/202081

Remember me

Oh okay .... Let me try proving theorems myself
I have been getting some success

Soham Chowdhury

Balarka voice That's very good.

Abdou Abdou

is it the right limit ? or is there some tricky way i was not been aware of ?

Soham Chowdhury

13:03

wait.

@Balarka, does one get $SO_3$ if one sets $A = -A$ in $SU(2)$?

If so, I think I'm beginning to "get" quotients.

Remember me

sulphur trioxide?

Soham Chowdhury

Rotation group SO(3)

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance (so it is an isometry), and orientation (i.e. handedness of space). Every non-trivial rotation is determined by its axis of rotation (a line through the origin) and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity...

Remember me

Topology?

Altop?

Soham Chowdhury

Na re.

Just groups.

$user image$

Remember me

Oh.... DF has this??

Soham Chowdhury

13:07

Aluffi ^_^

Remember me

gtg cming back in 5min

skill patrol

later pal

Remember me

@Soham How do you get those small shots

hey@Skill

Soham Chowdhury

@Balarka DF had around four examples of quotients. Am I missing something?

Remember me

Yes ! Yes I have understood first half of your blog @PaulP

Chris's sis the artist

13:29

@robjohn I just evaluated another amazing series!!!!!!!!!!!11

$$\sum_{n=1}^{\infty} \frac{H_n}{n^2}\left(\zeta(2)-1-\frac{1}{2^2}-\cdots-\frac{1}{n^2}\right)$$

Great!!!!!!!!!!!!!!!!!!1 Well, I simply love myself (for getting such results)! :-)

Soham Chowdhury

You've been doing this $\zeta(2) - \left(\sum_{k=1}^n \frac{1}{k^2}\right)$ thing for a while.

Agawa001

whats ζ(2) ?

Chris's sis the artist

@SohamChowdhury That can be written in terms of polygamma, but I like it more the way it looks in this form.

Soham Chowdhury

@Agawa001 $$\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s}$$

Agawa001

ah ok

?

Soham Chowdhury

13:35

That's how it should look.

Agawa001

u r editin it too much !!

ok

Soham Chowdhury

u hv prblm? i dnt k???????

That's painful :P

Agawa001

i was just bein confused

the actual one is correct

i assume that ζ(1)=e ?

m i wrong ?

yes i think

robjohn

@Chris'ssistheartist That looks like something we've already done, but I could be mistaken.

Chris's sis the artist

@robjohn No, definitely no. That one is very advanced amongst series (especially when only using elementary tools of real analysis).

Balarka Sen

13:42

@SohamChowdhury I know Q_8 is not the only one

Remember me

Hello@Balarka

Chris's sis the artist

@robjohn We possibly discussed this version $$\sum_{n=1}^{\infty} \frac{H_n}{n}\left(\zeta(2)-1-\frac{1}{2^2}-\cdots-\frac{1}{n^2}\right)$$ that I can presently compute in 3 different ways. I showed you in the past my solution to it. I think r9m added it to his blog.

Balarka Sen

hi

@SohamChowdhury really? oh well.

come up with your own examples, then, I guess :)

Soham Chowdhury

hey.

$\Bbb{R/Z} = S^1$?

I think I saw that in an exercise.

Balarka Sen

R/Z has two different intepretations

Gato

13:45

Hi

Balarka Sen

one is the standard quotient group thing

Soham Chowdhury

@Balarka, does one get SO3 if one sets A=−A in SU(2)?

I pinged you with that.

Balarka Sen

i guess

Soham Chowdhury

woo.

Gato

@BalarkaSen It's not $\Bbb{R}/2\pi\Bbb{Z}$ ?

Balarka Sen

13:46

same thing

Soham Chowdhury

I think I'm beginning to grok quotients :)

Chris's sis the artist

$$\large \text{Thank you Lord for the brilliance!!! :-)}$$

Soham Chowdhury

we have a god-fearing lady who is adept in the sciences here, I see. :P

The Artist

@Chris'ssistheartist have u ever tried/worked on to solve the puzzles of cicada 3301 ? :)

Soham Chowdhury

@BalarkaSen ki korli?

Chris's sis the artist

13:47

@TheArtist No, never heard of them. :-)

Gato

@BalarkaSen For any group generated by a single element, the order of the group is the order of the element ?

Balarka Sen

@SohamChowdhury what d'you mean?

@Gato yes

Soham Chowdhury

what did you do all this while?

The Artist

@Chris'ssistheartist you should check them out , its pretty cool :)

Balarka Sen

@SohamChowdhury schoolwork

Soham Chowdhury

13:48

i have to start. procrastinating like anything, but there's only a week left.

when does yours reopen?

Balarka Sen

13th

or 15th, I dunno

Soham Chowdhury

15th-e extend korlo toh.

Balarka Sen

yeah, heard about that

Soham Chowdhury

Ours was that from the start too. 1 week :(

Balarka Sen

are you in eng mid or beng mid?

Soham Chowdhury

13:51

eng. you?

Balarka Sen

beng.

Soham Chowdhury

maybe

@BalarkaSen forgive me for not knowing, but for how many subjects do you have to appear for exams?

Balarka Sen

7

Chris's sis the artist

@TheArtist Thanks! :-) I have to finish a book first. :P

Soham Chowdhury

@BalarkaSen physical science is nice, I guess.

The Artist

13:52

@Chris'ssistheartist heheh ok :) oh yes, hows that work going ?

Balarka Sen

sure is, @Soham

Soham Chowdhury

I heard ICSE is going to combine PCB.

we missed it.

anyway.

when do you wake up in the morning?

Balarka Sen

WBBSE guys after this batch are going to have open-book exams

Chris's sis the artist

@TheArtist I'm amazed of myself every day :-)))))))) God really created us very powerful beings. All is fine, I have a lot of marvellous work. :-)

Soham Chowdhury

@Chris'ssistheartist God, if he exists, sure has an eye for mathematical beauty.

3

Chris's sis the artist

13:54

@SohamChowdhury Well-said!

Balarka Sen

I think whoever starred that interpreted it differently.

Soham Chowdhury

But Occam's razor is awesome. @Balarka, yes, I guessed.

when do you wake up?

Balarka Sen

sigh. 9 AM. happy?

Chris's sis the artist

@SohamChowdhury He exists 100%.

Soham Chowdhury

Let's not argue. :)

The Artist

13:56

@Chris'ssistheartist :) very glad you feel like tat, yes very true :) i wish i felt the same about myself everyday

Agawa001

thats math room theological debates are discussed elsewhere

Soham Chowdhury

@BalarkaSen yes :P

The Artist

@Chris'ssistheartist agreed with you 100% :) I believe that God exists too

Gato

@BalarkaSen Wen I look at the canonical surjection between Z onto Z/nZ : $\pi$, as $\pi$ is surjective, then a subgroup of $Z/nZ : K=\pi(\pi^{-1}(K))$. As $\pi^{-1}(K)$ is a subgroup of $Z$ then it looks like $kZ$, therefore $K$ is generated by $\pi(k)$. Now if I denote $\overline{k}=\pi(k)$, I have $\vert K\vert=\vert \overline{k}\vert$, then $\pi(dk)=d\overline{k}$ where $d$ is the smallest integer such that $d\overline{k}=\overline{0}$. How can I finish to prove that $d=n/k$ ?

Soham Chowdhury

@BalarkaSen basically mine shifted three hours.

people, cut the starring

Balarka Sen

13:57

we're having a star-war.

2

Soham Chowdhury

couldn't resist.

Agawa001

lol ^^

Soham Chowdhury

i'm a filthy hypocrite sometimes.

skill patrol

what does "exist" mean when you are talking about God?

Balarka Sen

@skill "believe it or I'll punch you"?

@Gato OK, what d'you want to prove, precisely?

Gato

13:59

@skillpatrol "exist" about what ?

skill patrol

god

Gato

@BalarkaSen That for any cyclic group there is a unique subgroup of that form with order $n/k$

Transcript for

$Mathematics$ Mathematics