Another math problem! (Not HW)

[Max™]

Abelian Group -- from Wolfram MathWorld

All cyclic groups are abelian, infinite or not, so it shouldn't be any different from the ring of integers under addition.

[joft]

you will kick yourself about how obvious it is: just write down the isomorphism

say your infinite cyclic group is generated by x
map f(x) = 1
check each:
1) well-defined
if a = b then a = x^n implies b = x^n (otherwise you get some finite order)
so f(a) = n = f(b)

2) a homomorphism
say a = x^n, b = x^m, then f(ab) = f(x^{m+n}) = n + m = f(a) + f(b)

3) surjective
for any integer n, f(x^n) = n

4) injective
suppose f(a) = f(b) then a = x^n and b = x^m for some n, m, and
f(ab^{-1}) = f(x^{n-m}) = 0 so n-m = 0 and a = b

[Max™]

Yeah, I was kinda confused too, I was looking at it trying to figure out what he meant a few times.

I saw infinite cyclic and integers, and I was like "isn't that just an extension of the abelian arguments..."... maybe I wasn't as confused as I thought.

[Woozie]

Holy crap that was easy. I didn't know where to start because I couldn't figure out how to make a mapping from an unknown group to Z. I didn't think to define the mapping in terms of the generator @_@

Edit: You're right Max, all cyclic groups are abelian, but that doesn't mean I can apply the fundamental theorem of finite abelian groups to a cyclic group with infinite order. That's why I said the site you posted wasn't sufficient.

Edit 2: Now that I think about it, I once had to prove that Z has only two automorphisms. The proof you posted is very similar to that proof because it relies on the fact that generators have to be mapped to generators, and that once you have a map from a generator to a generator in Z, the mapping of every other element automatically follows. For some reason I didn't think back to that, but it seems so obvious now x_x

[joft]

the finite theorem extends: Finitely generated abelian group - Wikipedia, the free encyclopedia

[Woozie]

Hmm, interesting. I don't really understand it that well. I'm not seeing how to connect this and the fundamental theorem of abelian groups so that I can apply it to Z. and I can't find it in any of my old group theory books.

Edit: Nevermind, I just found it in one of my books. I'll read up on it. Thanks, joft.

Edit 2: I hope you're still around tomorrow, because I can already tell that if I have enough free time to read through this, I'm going to have a lot of questions @_@

I'm taking a course next semester on Special Topics in Group theory, so I'm trying to brush up on my group theory this month and the next. I'm not exactly sure what he expects us to remember (or if he'll review the stuff first).

[Max™]

T.T

Thank him but not me, I was bitching about the abelian argument for a reason.

He was better at explaining it though.

I was like "isn't that it, right there?" in my head.

They're useful in GR, main reason it came to mind.

[Sylvrdragon]

Jesus you guys are nerds...

Spoiler: show

I want in ; ;

[Woozie]

T.T

Thank him but not me, I was bitching about the abelian argument for a reason.

He was better at explaining it though.

I was like "isn't that it, right there?" in my head.

They're useful in GR, main reason it came to mind.

My bad Max, I appreciate your help too. If it wasn't for you, I probably wouldn't have realized there was a gap in my knowledge somewhere that could affect me in class next semester. So I guess I should be thanking both of you and not just joft.

[Julian]

I'M BACK EARLY, I WONDER WHAT THAT MEANS?

sounds weird, I can't help but think it's going to be some stupid trick like using the 1-norm instead of the 2-norm

the only actual interesting possibility i can imagine would be something like
Cantor function - Wikipedia, the free encyclopedia

except that thing has arc length 2

yeah any path between (0,0) and (1,1) is, by definition, at least as long as the shortest path-- the geodesic with length sqrt2

I love you.

Anyway, story time:

If anyone frequents the irl pic thread, you'll know there's a guy that looks JUST LIKE TSUKO in my physics lab class. Except, the guy is a gigantic faggot. (First year physics major, taking physics 2 first semester, thinks he's the fucking shit, but he's just an asshole)

Anyway, TA gets there, asks if anyone got an answer, me and him go up to him.

Tsuko-look-a-like decided to be a smartass and just redo the measurements, and then the TA went NO, I said x-y plane, and apparently he set the distance of that shortest line to root 2 to set the coordinates.

So, I had on one side of my paper the cantor function, and the other side a whole bunch of ideas that you came up with. I showed him the other side, he glanced at it, and then said "Did anyone come up with the Cantor function?"

And I was like, oh, shit, so I flipped my paper over and went "Oh, I did I did I did!!!!11 ^_____angelkitty______^" and showed it to him. Tsuko-look-a-like, at this point, got pissed, and insisted that he knew what the Cantor function was, and spent the next few minutes trying to explain that he knew what it was.

I don't know if he got his points, because the TA gave me a 20/20 and let me leave.

FUCK YEAH!

Thanks everyone <3

[ozz]

i think you're in my physics discussion

[joft]

lol wtf, cantor function has arc length = 2

[Max™]

Yeah... I'm confused there on the Cantor being what he wanted.

Just returning the favor Woozinator, for all the random math I do know, you pointed out how weak I was in other areas, so thanks as well. :D

[Julian]

i think you're in my physics discussion

wat

You mean like, regular physics discussion? Although I don't know how this topic about my physics lab has anything to do with the physics discussion but... thursdays, with some indian guy with a french accent that bribes us to answer questions with extra credit? <_<

[Woozie]

lol wtf, cantor function has arc length = 2

Don't question it. You just got Julian out of one of his labs with 20 free points and that's what's important lol

By the way, I talked to a group theory professor today and he doesn't seem to see the connection between finitely generated groups and showing that Z is isomorphic to every infinite cyclic group. It's not that big of a deal though, since you already posted a different proof.

[Woozie]

Jesus you guys are nerds...

Spoiler: show

I want in ; ;

Give it a few years of studying the books and subjects we posted. Depending on how fast you go, it wont be *too* long.

The biggest problem with teaching myself stuff is that I have a tendency to want to do *every* single problem in the book to make sure I understand it. When you have a teacher, you have homework and tests and a professional mathematician to gauge your understanding. When you're on your own, you pretty much have to just do every single problem, and that's very time consuming.

To be honest, the math stuff wasn't too difficult for me. Physics is where I have the most trouble. I don't know why I'm even persuing a career in physics. I'm much better at math than physics, and by the time I get my bachelors in physics, I'll have my masters in math (so I'd be closer to a PhD in math than in physics). I honestly don't want to have to choose between the two. I wish I could get a PhD in both.

[Ramor]

physics is applied math, somehow i think if anyone's that interested in a subject so heavily rooted in math (which is something they at least like) they won't have any trouble with it, especially considering how proficient you seem to be at all of it woozie.

teaching yourself math isn't so bad, i'd say try to do as much as you can without a calculator though, the easiest way to be really good at math is to know how to do things without a calculator you'll learn a surprisingly large amount just from doing everything out by hand. even knowing how to do sin(75) without a calculator just because you can do sin(30+45) = sin(30)cos(45) + cos(30)sin(45), stuff like that has helped me understand a lot of things much easier because i have a greater understanding of the tools people had available to them when they were first figuring it all out

[joft]

yeah I wasn't trying to make a connection. in fact, I'd say the proof you wanted is sort of a lemma necessary to prove the finitely generated theorem. I just brought it up as an interesting and related fact, and it shows that the classification Max brought up actually is relevant to your question (but with the implications reversed)

I'm currently (just a first year) phd math student. I would say to anyone who's interested in math/physics enough to consider doing a phd in them to remember that you can always study them as a hobby, and that there are some advantages to doing it that way (no grades or exams, it's much more fun when you know you're doing it for leisure, you get to pick the topics and the textbooks so you can choose books with the highest readability, etc). there are also numerous advantages to doing the phd-- chief among them accreditation for your knowledge. but one thing that is definitely not an advantage is employability. (edit: ok actually your employability will be marginally better with a phd degree,
but that margin alone is definitely not worth the amount of effort between master's and full phd)

there's a cold, hard truth, that academia trains orders of magnitude more people than it needs to replenish its own ranks. and a phd in a non-professional program mostly trains you to be an academic. I was naive when I applied to phd programs, thinking that going to a top 20 program would make me a good candidate for academic jobs. and the truth is I probably could find a job in academia after graduating, but it would mostly be teaching with little or no research and probably wouldn't be tenure track. I've decided to pick a specialization with employment prospects outside of academia, like numerical analysis (basically anything applied with lots of emphasis on using computers). so not only will I have job options outside of academia, but my options within academia will be better because of supply and demand.

a lot of people give similar warnings to this and then go on to say "you need to be ABSOLUTELY SURE about going in to academia before you do it, because it's SRSBSNS." but the truth is, if you get funding, it's not a bad lifestyle, and it does provide temporary shelter from getting a real job-- which won't happen anyway if you're graduating now. I'd say the only people who really need to take the decision seriously are people who have some other options that are really good, and which they might no longer have a few years from now if/when they drop out of a grad program.

and if you think it would make you bitter to work in a job that isn't related to your phd training you also might want to reconsider. because you can't expect that you will become a professor. I believe there's a bigger trimming between phds -> professorship than there is between bachelors -> grad school.

edit: also, even if you do become a professor, there's a very big likelihood that most of your time will be spent teaching stuff far below your level of training. very few professors work in top research institutions, and if you aren't doing that then most of your time will be spent teaching calculus or maybe even algebra or arithmetic. so even if you become a professor you are still highly unlikely to use your actual training as an academic or even teach classes related your subfield of specialization

[Eliseos]

physics is applied math, somehow i think if anyone's that interested in a subject so heavily rooted in math (which is something they at least like) they won't have any trouble with it, especially considering how proficient you seem to be at all of it woozie.

teaching yourself math isn't so bad, i'd say try to do as much as you can without a calculator though, the easiest way to be really good at math is to know how to do things without a calculator you'll learn a surprisingly large amount just from doing everything out by hand. even knowing how to do sin(75) without a calculator just because you can do sin(30+45) = sin(30)cos(45) + cos(30)sin(45), stuff like that has helped me understand a lot of things much easier because i have a greater understanding of the tools people had available to them when they were first figuring it all out

I cannot live without my TI-89 Titanium. I cannot do simple math to save my life, I generally only use it for plugging in numbers to make sure I didn't make some silly mistake (which often happens). The automatic unit conversions and the constants it has are badass, it even has the Rydberg Infinity Constant.

[ozz]

wat

You mean like, regular physics discussion? Although I don't know how this topic about my physics lab has anything to do with the physics discussion but... thursdays, with some indian guy with a french accent that bribes us to answer questions with extra credit? <_<

yeah, the guy who doesnt change clothes@3pm