Mathematics - 2024-04-24 (page 2 of 2)

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Obliv

6:12 PM

I'm not seeing how $G \cong N_1\times N_2\times \dots$

$f(a_1,a_2,...) = a_1\cdot a_2\cdot \dots $ doesn't seem injective

for example $\mathbb{Z}_{12}$, $(0,1)\mapsto 1$ and $(1,0)\mapsto 1$ for "addition"

Thorgott

that needs a lot more context to make sense

Obliv

yeah and I just realized why it's injective

they're disjoint normal subgroups of $G$ so you can't write $(a,b) \mapsto c$ and $(b,a)\mapsto c$ because $b \notin A$, $a\notin B$ or whatever

i'm still not a fan of using $=$ to mean $\cong$

it's just like the addition near-ring thing.. I have a preconceived notion of $=$ and now they're using it and $\cong$ interchangeably T_T

Thorgott

6:30 PM

typically, "=" means "natural isomorphism" or at the very least "an implicit canonical choice of isomorphism", you typically wouldn't use it to denote an arbitrary isomorphism

Obliv

natural isomorphism as in between groups?

I forgot what that means, it was in dummit & foote

I think it was $\pi$ to denote natural homomorphism

Thorgott

it's not something you should worry about

Xander Henderson

@Obliv Gross. $\iota$ is for isomorphisms, $\pi$ is for projections.

Obliv

why ioughta

shakes fist

Xander Henderson

I was taught to pronounce $\iota$ as 'YO-tah', so it took me a second to cotton on to what you were punning there.

Obliv

6:37 PM

what is this called again: $gcd(a,b) = 1 \implies d = au+bv$

does it have some fancy name

Xander Henderson

It's some guy's theorem.

Obliv

ok I will credit some guy then

Xander Henderson

B something...

Not Binet... that's for the Fibonacci sequence...

BEZOUT!

That's the guy's name, I think.

Bézout's identity

In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout who proved it for polynomials, is the following theorem: Here the greatest common divisor of 0 and 0 is taken to be 0. The integers x and y are called Bézout coefficients for (a, b); they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that |x| ≤ |b/d| and |y| ≤ |a/d|; equality occurs only if one of a and b is a multiple of the other. As an example, the greatest common divisor of 15 and...

Obliv

Thanks bezout

Thorgott

$\iota$ is for structure maps into a colimit, $\pi$ is for structure maps out of a limit

Xander Henderson

6:42 PM

Obliv

I keep wanting to switch between add. and mult. notation for groups even though the group is under addition D:

Xander Henderson

It's a group. What does it mean to be "additive" or "multiplicative"? There is one operation.

Obliv

I gotta remember that half of having the stuffs in math is having swag & writing wicked nice

like Ted's blackboard writing

Xander Henderson

Meh.

Obliv

being clearly legible & aesthetic earns you 10 leslie coin

or it should

@XanderHenderson Wait isn't it possible to have a set be a group under diff. operations though

like ($\mathbb{R},+$) and ($\mathbb{R},\cdot$)

Xander Henderson

6:48 PM

@Obliv Sure. But unless you are specifically thinking about some set of numbers, the different operations are just abstractions, and would probably be denoted $+_1$ and $+_2$ (if you needed to distinguish them).

Obliv

yeah I guess you're right. It's not like $\mathbb{Z}_n$ is a group under multiplication anyway. but having consistency in notation is important imo

everything is blurring together

is $U_{n}$ in general all the elements in $\mathbb{Z}_n\setminus \{0\}$ that are relatively prime with $n$

yesh

Xander Henderson

@Obliv I mean, maybe? I've not seen it in any generalized setting to mean that, but perhaps your book / professor uses it that way?

By the way, $\mathbb{Z}_n$ isn't great notation. Better to write $\mathbb{Z}/n\mathbb{Z}$ (or $\mathbb{Z}/n$, if'n yer lazy).

Obliv

oh sorry I should have said $U_n$ to mean units

yeah that makes sense, to denote quotient group

Xander Henderson