Mathematics - 2017-10-13 (page 2 of 7)

Eric Silva

01:00

i thought you were asking it with incredulity

Leaky Nun

@MatheiBoulomenos tell us

Faust

:s

MatheiBoulomenos

Okay, but the whole argument is quite long

Faust

i dont understand that word eithier

Leaky Nun

@MatheiBoulomenos it doesn't matter

Eric Silva

01:00

it means when you are unwilling to like accept something

mdave16

what are we talking about?

Balarka Sen

@Forever I should watch Blade Runner

Faust

Im trying to acertain if its the type of beard he has or the method to which he lets his beard grow that you admire.

like lots of beer?

Balarka Sen

I'm too far down in obscure shit that I haven't even looked at classics

Faust

lots of couch sitting?

or just how he trims it?

Eric Silva

01:02

he just grows a nice full beard and also is an attractive dude who looks good with a beard

Faust

ah less intresting :P

mdave16

@BalarkaSen, you may enjoy The Lobster and also Eternal Sunshine Of The Spotless Mind

MatheiBoulomenos

Alright, here is a proof that there is no simple group $G$ of order $33264=2^4\times3^3\times7\times22$.
Consider the number of $N_{11}$ $11$-Sylows, by Sylow we have $N_{11}=1,12,56,144$.
Case 0: If $N_{11}=1$ we're done.
Case 1: If $N_{11}=12$, then the normalizer of a $11$-Sylow contains an element of order $77$,
but as $G$ is simple and acts non-trivially on the set of $11$-Sylows by conjugation, we have an embedding $G \to S_{12}$, but $S_{12}$ does not contain any element of order $77$.
Case 2: If $N_{11}=144$, then the normalizer of a fixed $11$-Sylow $P_{11}$ has $3\times7\times11$ …

Eric Silva

eternal sunshine of the spotless mind is such an emotional trip

MatheiBoulomenos

Case 0 and Case 1 are not problematic

I'm not sure if the argument from Case 2 and Case 3 works

I mean I can't see why not

But it seems like a strange argument

Faust

01:05

@MatheiBoulomenos is that Galois GT or something?

Leaky Nun

@MatheiBoulomenos 11, not 22

MatheiBoulomenos

that's pure group theory

Balarka Sen

@mdave16 I have been recommended Lobster, damn you all, you're just reminding me how much movies are there on my queue list but i haven't watched yet

Leaky Nun

and you actually did it lol @MatheiBoulomenos

Balarka Sen

actually both Blade Runner and Lobster are reco'd by @Alessandro

Leaky Nun

01:06

@Faust technically, this is Sylow theory

Faust

@MatheiBoulomenos when do i get to learn it?

mdave16

I second the recommendations, I used to work in a cinema, so i've seen a fair few

Leaky Nun

@Faust in any decent group theoretic course

definitely way before you even touch Galois

Faust

we already moved on to rings today

MatheiBoulomenos

@LeakyNun where did I write 22?

Faust

01:07

so no no no

Balarka Sen

@mdave16 woo then you should hang around more often

Leaky Nun

@MatheiBoulomenos on the first line apparently

Wheat Wizard

@TedShifrin The reason I missed the glaring error is because it only renders in the review queue. If you look at the markdown the error is in fact not there. I thought I had been going crazy.

Leaky Nun

@Faust in my book Sylow comes after rings

Balarka Sen

I actually didn't super like Eternal Sunshine

Wheat Wizard

01:07

Must be a bug or something

MatheiBoulomenos

@LeakyNun ah right

Faust

@KasmirKhaan havent seen u in awhile hope ur doing good

MatheiBoulomenos

too late to fix it

Balarka Sen

my tastes are weird, hipsterish

Ted Shifrin

It sure looks like the sentence is inside the dollar signs @Wheat

mdave16

01:07

yes, the lobster will be relatable then

Leaky Nun

@MatheiBoulomenos you just need to do Ctrl+F followed by two hits of "2"

Irregular User

https://math.stackexchange.com/questions/1475079/every-point-lies-on-a-unique-secant-through-c

Anyone know why "So one must have Pi=Qj for certain i,j" in the answer is true?

Balarka Sen

@mdave16 Great I'll definitely put it on top of my list

Eric Silva

@Balarka ud probably like my roommates student films tbh

mdave16

what else is on the list?

MatheiBoulomenos

01:08

@LeakyNun I didn't think that you could search rendered LaTeX lol

But I guess it works since it's only rendered by a script

mdave16

I feel almost like we're in a small pub and all the conversations crossover each other... but at the same time, i don't feel rude

Leaky Nun

@MatheiBoulomenos I won't go into the technical details (of mathjax)

Wheat Wizard

@TedShifrin Its between two other statements $\sup S $ can only be equal to $\sqrt{29}$, it doesn't render on the actual question.

Leaky Nun

but could you explain how we know that the normalizer of a 11-Sylow contains an element of order 77 in case 1? @MatheiBoulomenos

Balarka Sen

@mdave16 Hahah nice description, I like it

Faust

01:10

Prove that any positive integer of the form n = (6k + 1)(12k + 1)(18k + 1), where all
three factors are prime, is a Carmichael number. ( A Carmichael number is one such that $a^c \equiv 1 $ for all bases c s.t (a,c)=1 anyone know how to do this?

Balarka Sen

so, on my list, there's this indie movie Primer and Upstream Colors by the same director

Ted Shifrin

Hmm, I don't understand the discrepancy, @Wheat, but I've put more time into this than I care to.

Wheat Wizard

Ok fair enough

It renders on the question so I'm happy

Thanks for everything

mdave16

@BalarkaSen, others to add are Mood Indigo and The Zero Theorem. Anyway, it's quite late here and i have an interview tomo, night all

MatheiBoulomenos

@LeakyNun sure, we know that the 11-Sylow subgroup is a normal subgroup of its normalizer. If we compute the order of the normalizer, then we see that it has an element of order 7 by Cauchy. Now there's a lemma in elementary group theory that says that if you have a normal subgroup and any subgroup, then you can take the product of that and it will be a subgroup again. So the normalizer contains a subgroup of order $77$. Groups of order $77$ are cyclic by Chinese Remainder Sylow

Faust

01:12

@TedShifrin can u reccomend a decent abstract algerbra book for me?

Balarka Sen

There's a bunch of horror movie that I have watched but didn't tick off

@mdave16 Ohh nice

but yeah g'night

MatheiBoulomenos

I just made these kind of arguments so often for myself that I don't bother to write them down anymore

Leaky Nun

@MatheiBoulomenos what is "Chinese Remainder Sylow"

Faust

Gnight balarka

Balarka Sen

I'm still here :)

Ted Shifrin

01:13

@Faust: Although I don't love Fraleigh, I think it's a decent level of exposition for you.

Faust

Morning @BalarkaSen ?

Balarka Sen

ye

Faust

ill see if my libary has it

Leaky Nun

@MatheiBoulomenos why does $G$ admit a faithful group action to the set of the 12 Sylow subgroups?

MatheiBoulomenos

@LeakyNun you use Sylow to show that the Sylow subgroups are normal (and they are cyclic due to their order) and then you use CRT to conclude that the whole thing is cyclic

Leaky Nun

01:14

@MatheiBoulomenos lol ok

MatheiBoulomenos

Any nontrivial group action of a simple group is faithful

as the kernel is a normal subgroup

Leaky Nun

why is it non-trivial?

and how do you compute the order of the normalizer?

MatheiBoulomenos

One of the Sylow theorems say that the action by conjugation is transitive

the index of the normalizer is equal to the number of Sylow subgroups

(this is an application of orbit-stabilizer)

Ted Shifrin

I have always loved that equality. It came up not too long ago.

Leaky Nun

but transitive doesn't mean faithful? @MatheiBoulomenos

Faust

01:17

Nvm i got the carmichael number question

MatheiBoulomenos

transitive certainly means non-trivial

and non-trivial together with the simplicity of the group implies faithful

Leaky Nun

oh

@MatheiBoulomenos so you mean x/11 = 12?

MatheiBoulomenos

@TedShifrin would you mind checking my argument? I'm really unsure and would like to know if it works

Ted Shifrin

Nah. I'm about to go cook dinner, and my back is hurting a lot, so I don't have energy to concentrate on stuff that I don't think about every day.

MatheiBoulomenos

okay, np

Maybe I'll post it as a question on the main site

@LeakyNun you just divide the order of the group by the number of Sylow groups

that gives you the order of the normalizer

Leaky Nun

01:20

whaaat

I've never heard of this

MatheiBoulomenos

I thought you learned Sylow theory?

That's pretty useful in applying Sylow theory

Faust

If$ a, p \in \mathbb{N}$ and $a^{p}-1 $is prime, then a = 2 or p = 1 i can show a>2 and p>1 implies $a^{p}-1 $ is not prime always but dont know how to finish the argument?

Leaky Nun

I only know that p|[N(H):H] @MatheiBoulomenos

MrAP

People please help me solve this question:The three different sections of a library need the services of 3, 4 and 5 workers respectively. If 12 workers are available, in how many ways can they be allotted to different sections.

Leaky Nun

@Faust contrapositive

the icons of @MrAP and @MatheiBoulomenos look too similar

Faust

01:22

i dont know what the contrapositice is O.o lemme google it

Leaky Nun

the only way I can distinguish is by the difficulty of the things they say

Faust

lol

MrAP

Haha

Ted Shifrin

@Faust: Converse and contrapositive you need to know in your sleep.

Mathei has arrows.

Faust

i know they exist

and i know converse

a if b converse is b if a?

Leaky Nun

01:23

yes

MatheiBoulomenos

@LeakyNun Well, it's really just the Sylow theorem that says that two $p$ Sylow subgroups are conjugate. You take the action of the group on the set of its $p$ Sylow subgroup and apply Orbit-Stabilizer. Note that the stabilizer of one $p$ Sylow group under this action is its normalizer

Faust

its like p imply q is equiv to not q imply not p is equiv truth value?

MrAP

I have an exam in about 1 hour.

Faust

Does anyone know why i picked math? physics is so much easier O.o

Ted Shifrin

Actually, upper-level physics is not very easy.

MatheiBoulomenos

01:26

I find physics much more difficult

Leaky Nun

@MatheiBoulomenos btw maybe you could help me on this: groups and rings and fields I can still somewhat visualize, I can understand what's going on behind the lines, but I completely lost it when I come to Sylow theory: those are just lines of texts and formulas for me, and I can't really "understand" the proofs. They became facts that I just memorize, unlike other things that I really understand and can visualize in my head.

Faust

how do u write not n latex?

Leaky Nun

@Faust \neg

equiv means equivalent

Ted Shifrin

or \not

depends what you're doing

Faust

how do u write imply?

Leaky Nun

01:26

you're using the wrong command if you need to use \not @TedShifrin

Faust

sorry im retarded today

Leaky Nun

@Faust \implies

Faust

been doing math since

5am

and its 626pm

Leaky Nun

@Faust go rest

Faust

$ p \implies q $ is equivalent in truth value to $ \neg q \implies \neg p $

Leaky Nun

01:27

@Faust I said \neg

Faust

hmm that looks broken

MatheiBoulomenos

@LeakyNun well, I don't visualize the Sylow theorems directly. I see them as a demonstration of the fact that group actions can tell us a lot about groups. If you look at the proofs, then they are mostly group actions. I just have a bit of experience applying them, due to my hobby of Sylowing that I mentioned earlier

Leaky Nun

@MatheiBoulomenos I envy the hobby of Sylowing lol

Faust

less broken? is that the contrapositive?

Leaky Nun

but well I'm two years behind you, I can really develop it now

@Faust yes

MrAP

01:28

Help me somebody

Leaky Nun

@MatheiBoulomenos the problem is I can't visualize group actions / normalizer / conjugate / etc

I just view them as words with no meaning

Faust

@MrAP post it again ill take a whack at it

MatheiBoulomenos

@LeakyNun group actions are how I think about groups

Faust

@LeakyNun that solves it for me. also you never did answer me this morning

Leaky Nun

@MrAP if you have an exam in an hour, it is not going to help you to know the answer to that one specific question that won't appear in the exams

@Faust what did you ask me?

MatheiBoulomenos

01:29

and I think about conjugate like a change of basis in linear algebra

Faust

@LeakyNun how old are you

Ted Shifrin

@MrAP: How many ways can you pick a group of 3 people from 12?

Leaky Nun

I am 18 @Faust

Faust

@TedShifrin oh oh i know that one!

Leaky Nun

@MatheiBoulomenos "I can't visualize group actions"

@Faust of course you know

but MrAP doesn't know

Faust

01:30

shit your a spring chicken

MatheiBoulomenos

@LeakyNun well, maybe look at some examples. $S_n$ acts on an $n$ element set, $D_n$ acts on a regular n-gon etc.

Leaky Nun

@MatheiBoulomenos I can visualize those

Ted Shifrin

I told you to think of symmetries of geometric figures, Leaky, and you rebuffed me.

Faust

also FYI everyone 3300 divide by 2 is not 1500.

MatheiBoulomenos

$\operatorname{GL}_n(K)$ acts on $K^n$

Leaky Nun

01:32

@TedShifrin sorry :c

@MatheiBoulomenos what is the relationship between normalizer and centralizer?

Faust

Hopefully my prof dont take too many marks off of my midterm for that.

MatheiBoulomenos

@LeakyNun the centralizer is a subgroup of the normalizer

Leaky Nun

@MatheiBoulomenos in terms of group action?

Ted Shifrin

@MrAP: If you can't answer my question in a few seconds, you're in big trouble.

Leaky Nun

@TedShifrin in exam, that is

in life, not really a big trouble

Ted Shifrin

01:34

He isn't here about life.

MrAP

Oh i am sorry.

I was inactive

Leaky Nun

@MrAP you can be friends with @PVAL-inactive

Faust

@MrAP the answer is one

MrAP

just a moment please

Ted Shifrin

smacks Leaky

3

Leaky Nun

01:35

@TedShifrin :D

PVAL-inactive

?

MrAP

12C3

Leaky Nun

@PVAL-inactive LOL

MrAP

.

Faust

lol

MatheiBoulomenos

01:35

@LeakyNun the normalizer of a subgroup acts on that subgroup by conjugation (and it is the largest subgroup of the ambient group so that this action is well defined), this action is actually a group homomorphism $N_G(H) \to \operatorname{Aut}(H)$, the kernel of this homomorphism is the centralizer

Ted Shifrin

OK, @MrAP. And now how many are left? We have to next pick a group of 4.

Leaky Nun

@MatheiBoulomenos this kind of make sense; why have I never heard of it?

PVAL-inactive

@Ted You saw the proof I linked?

Ted Shifrin

Yeah, @PVAL, but I didn't study it carefully.

I still have the tab open.

MrAP

Wait let me first post the question again. The three different sections of a library need the services of 3, 4 and 5 workers respectively. If 12 workers are available, in how many ways can they be allotted to different sections.

Leaky Nun

01:36

@MatheiBoulomenos well-defined?

PVAL-inactive

I misremembered it but it avoids talking about the openness of singular holomorphic functions.

MrAP

9C4?

MatheiBoulomenos

@LeakyNun yes, if you take an element that is not in the normalizer and conjugate the subgroup with that, you will get a different group

Ted Shifrin

OK. Now finish the question, @MrAP.

PVAL-inactive

It uses that polynomials are proper (which is still really easy to prove.).

Leaky Nun

01:37

@MatheiBoulomenos oh, closed lol

I was thinking of one elements mapping to two elements

Ted Shifrin

I am OK with properness of polynomials, @PVAL.

MrAP

Is it 12C3*9C4*5C5?

PVAL-inactive

I'm okay with the openness of singular holomorphic maps, but I'd rather use topology.

Ted Shifrin

(Sure. But of course 5C5 is 1.)

Meow Mix

howdy partners

Leaky Nun

01:38

@MatheiBoulomenos oh, and Aut(H) is a subgroup of S_H

Ted Shifrin

Hi Meow.

MatheiBoulomenos

yes

Leaky Nun

@TedShifrin next time introduce trinomial coefficients :P

Ted Shifrin

I taught those in my probability class, but I don't know if @MrAP knows them or not.

PVAL-inactive

@Meow I have a monopoly on the cowboy talk here.

MrAP

01:39

i dont'

Leaky Nun

@MatheiBoulomenos oh I learnt of this amazing fact that I'm not sure if you know: inner and outer automorphisms are called inner and outer because if you embed your group G to SG, an inner automorphism is a conjugation by an element inside G, and an outer automorphism is a conjugation by an element outside G (but in SG)

MrAP

don't

Meow Mix

too bad

Ted Shifrin

LOL @PVAL

Meow Mix

i already bought park place and boardwalk

MatheiBoulomenos

01:39

@LeakyNun that's really cool

Ted Shifrin

There aren't cowboys on them, @Meow.

Leaky Nun

@TedShifrin I'm just joking

Meow Mix

inner and outer automorphisms like inner and outer belly buttons?

Ted Shifrin

On that note, I'm going to cook dinner. Bye, y'all.

MatheiBoulomenos

Bye @Ted

Meow Mix

01:40

seeya tedulator

MrAP

Well thanks for the help.

Faust

gl with food

Let (sn) be a sequence of real numbers such that $ |s_{n+1} − s{n}| \leq \frac{1}
{2^n} for all n. Show that (sn) converges.

Leaky Nun

@MatheiBoulomenos "what is your hobby?" "taking random integers and proving the non-existence of simple groups of order thereof"

Faust

does this seem reasonable?

MatheiBoulomenos

@LeakyNun sure thing

Leaky Nun

01:43

@MatheiBoulomenos "you can start working next monday"

PVAL-inactive

@Faust that looks like the right idea.

Leaky Nun

@Faust work on your handwriting :P

Faust

the sum is wierd

i have dysgraphia

PVAL-inactive

Bound the tail s_n-s_m using a geometric series.

Faust

i did

i was worried though

the geometric sum i wrote looks wierd

Leaky Nun

01:44

@MatheiBoulomenos you're doing something wrong if you need to brute force the solution to $143=11a+7b$

seeing that $143$ is divisible by $11$

MatheiBoulomenos

I require that $a,b >0$

because else the order of the permutation is not $77$

I mean, it's faster to write one line of code than thinking about it

Leaky Nun

@MatheiBoulomenos so you just do 143 = 11x13 = 11x(7+6) = 11x6 + 7x11

wait, you solved it by code :o

which program?

MatheiBoulomenos

sure, but why is that the only solution

ghci

Leaky Nun

ic

case 2 is taking me everywhere lol

MatheiBoulomenos

what's ic?

Leaky Nun

01:47

I see

MatheiBoulomenos

ic

Faust

lol

Leaky Nun

@MatheiBoulomenos ubung macht den meister

MatheiBoulomenos

@LeakyNun that's for sure

Leaky Nun

@MatheiBoulomenos could you run me through case 2?

MatheiBoulomenos

01:48

@LeakyNun sure

We compute the order of the normalizer like I said

then we make the same argument to find an element of order $77$ in the normalizer

that's $x$

the next thing I want to show is that $x$ is not in the normalizer of any other $11$ Sylow subgroup

it's a consequence of the Sylow theorems that one Sylow subgroup is not contained in the normalizer of any other Sylow subgroup (for the same prime obviously)

so we note that $x^{7}$ is a generator of the Sylow subgroup

Max

I'm a bit stuck with the proof of Kruskals algorithm (that it produces a spanning tree)
> Let P be a connected, weighted graph and let Y be the subgraph of P produced by Kruskals algorithm. Y cannot have a cycle, being within one subtree and not between two different trees.
I don't understand the last sentence, what does ".. being within one subtree and not between two different trees .." mean?

MatheiBoulomenos

Now if $x$ would be in the normalizer of some other $11$-Sylow subgroup, so would be $x^7$

but that's impossible by the above

thus $x$ normalizes precisely one $11$-Sylow subgroup

Leaky Nun

how is it that Sylow came up with all these stuff a few hundred years ago and then everyone in the world is now learning about it lol

it amazes me that one man alone invented it

Faust

if n is a positive integer such that 24|(n + 1), then show that $ 24|\sigma(n)$ anyone know how to start this?

MatheiBoulomenos

Now comes the group action

$G$ acts on the set of Sylow subgroups by conjugation

Leaky Nun

01:54

@Faust if 24|(23+1) then show that 24|22?

Faust

sigma of n is the sum of all the divisors of n

Leaky Nun

oh

Faust

so in that case 24

cause 23 is prime

Leaky Nun

waaaat

$\Huge{23~\text{is}~\text{prime}~\text{:o}}$

Faust

the divisiors or 23 are 23 and 1

so the sum of those two numbers

Leaky Nun

01:56

that's like so amazing

Faust

is 23+1

O.o

i think your being sarcastic?

Leaky Nun

I always thought 23 = 5 x 7

Faust

now im fairly certain your being sarcastic

MatheiBoulomenos

Now this group action (which is faithful by the assumed simplicity of $G$) allows us to think of the elements of $G$ as permutations of the $11$ Sylow subgroup

Leaky Nun

@Faust hexadecimal

Faust

01:57

@LeakyNun nonsense

Leaky Nun

$23_{16} = 2 \times 16 + 3 = 32 + 3 = 35 = 5 \times 7$ :P

Faust

say base 16

Leaky Nun

hex-a-dec-i-mal.

MatheiBoulomenos

under this identification, an element of $G$ normalizes a $11$-Sylow subgroup if and only if the permutation fixes the "letter" that represents the $11$-Sylow subgroup

Leaky Nun

I sense some class equation thing going on

but alright, class equation is just orbits

1+11+...+11+7+...+7

MatheiBoulomenos

01:58

So we have just shown that $x$ normalizes exactly one $11$-Sylow subgroup

and we know the order of $x$

this allows us in this case by solving the equation $143=11a+7b$ to deduce the cycle type of $x$

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$Mathematics$ Mathematics