Today I reminded that the free product $G * H$ has $G$ and $H$ as normal subgroups.

Though, that raises a question... The free product $G * H$ has the direct product $G \times H$ as a factor group. What is the corresponding normal subgroup?

@DannyuNDos If $x, y$ are generators of free group on two elements then does $yx\in \langle x\rangle \cdot y$?

It doesn't because then $yx = x^ny$ for some $n\in\mathbb{Z}$ which leads to a contradiction

Huh, nevermind.

so $G = \langle x\rangle$ and $H = \langle y\rangle$ are not normal subgroups of $G*H$ their free product

The source of confusion was this: Given a group $G$ and its subgroup $H$, I tried to find the pushout of the identity map $\mathrm{id}: H \to H$ and the inclusion map $i: H \to G$.

I thought the result would be $(G * H) / H$.

Which is... isomorphic to $G / H$, in my second thought?

And still, $G * H$ should have $G$ and $H$ as factor groups.

Wait, $G / H$ isn't a thing; $H$ might be not normal. Dang.

pushing out along an identity map does nothing

D:

trying to decide if the following sentence is grammatical: What radius circle will a particle (mass m, charge q) follow if moving at speed v in a magnetic field B?

specifically the "what radius circle" bit

it seems grammatical to me, but i have no idea. and, there is sometimes a gap between grammatical and good exposition.

this is something ted could have answered, because when he went to school they still taught grammar

yeah, grammatical != "reads well"

(i'd hope that being grammatical is a necessary if not sufficient condition but i'm not confident of that)

part of the question is an implicit assertion, which is that a particle under those circumstances moves in a circle. if the point is that you don't want the student to prove/investigate/focus on that assertion, i might just state that fact - a particle doing bleh will move in a circle. now you, student, tell me the radius of that circle.

yeah, they're given the formula r=mv/qB on the equation sheet, and this is a multiple choice question

@Semiclassical it's not $\frac{mv^2}{qB}$?

in the last physics class i took, i think we did some kind of experiment relating to this equation to compute the mass of an electron.

no. mv^2/r = qvB, hence r=mv/qB

i'd say something about units but that would require me thinking about Tesla as a unit

ah, forgot that qvB was force

and life's too short for that

@robjohn ur mom is force!

Which branch does K-theory belong to, algebra or topology?

it began in algebraic geometry, in grothendieck's formulation of the riemann roch theorem, but topological K came pretty soon after

why not both

So it belongs to geometry then.

arxiv.org/abs/math/0012213

Too sadge geometry isn't broad enough to be considered the 4th branch.

Analysis, Algebra, and Topology are the three branches, but geometry?

@leslietownes you're pretty good at finding stuff on the internet. Are you a detective?

nope

@leslietownes do you have PhD??

where is all of this going. what do you really want to know?

@leslietownes your profession? But you're not telling me :(

i am a lawyer

i guess i forgot to answer that

@leslietownes so you have done math degree then you studied law?

yeah, why not

I think, you spend more time here than in court haha no offence :)

i don't know where you live or how the court system works there, but in the united states, it isn't exactly a badge of honor to have one's clients appearing in court all of the time

there's a good joke in the US about a country lawyer who says "i'm the greatest lawyer, every will i have written has been upheld by the supreme court"

the underlying premise of this joke is that you are f-ing up badly as an attorney if anything you do leads to a supreme court case

I can't find any wiki or nlab articles on tame sets and wild sets. Can someone suggest some references where I can find the definitions and basic stuffs about those?

i never saw such things, what field do they arise in?

3 manifolds. I don't know if they appear in other fields or not.

@SoumikMukherjee @BalarkaSen

Sounds like parts of what you study

topologically tame 3-manifold is a 3-manifold whose interior is homeomorphic to a compact 3-manifold. wild otherwise

I mean interior of a compact 3-manifold

@onepotatotwopotato does 3-manifold mean manifold with boundary here?

yes possibly nonempty boundary

@onepotatotwopotato Thanks

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