Jake1234

What are you asking vrouvrou? Is the equation true? Yes it looks like it.

@Vrouvrou please, does your nickname mean anything ? Just curious.

Vrouvrou does your nickname mean anything? I remember some french player of a game I used to play using it.

Hey

Or if it is defined, it's still just $V$, not isomorphic to the external direct sum of $V$ and $V$.

But for example you can take the subspace $V= \{ (0,0), (0,1) \}$ of $\mathbb{Z}_2 \times \mathbb{Z}_2$, take the external direct sum of $V$ and $V$, and you get something isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$, but there's not an internal direct sum defined for $V$ and $V$. (I think... might be nice if someone who understands this stuff checked it out xD.. I think I'll just ask someone in school to make sure).

for $\mathbb{Z}_6$, it's an internal direct sum of those two sets.

There's one in [direct sum]en.wikipedia.org/wiki/Direct_sum

I think there might be some slight differences depending on the writer, but it looks to me like: Internal sum of two sub-structures is defined for two sub structures that are 'independent', and it's all their combinations. External sum are the actual tuples from the cartesian product of the sub-structures. If the internal sum of two sub-structures is defined, it's isomorphic to the external sum, hence naming it 'internal direct sum'.

Internal product is actually what it 'generates' in the given space.

[title]math.ncku.edu.tw/~fjmliou/advcal/sumvspace.pdf I think this explains it for me well enough.

This is mostly for a basic course in ring/module theory

Yeah I understand it like that, just not completely sure I get it precisely enough.

Thanks, yeah I looked at that already.. sadly not really.

It says they're isomorphic, but are they the same set? It seems to me like they are. If I take the direct sum of $V_1$ and $V_2$, and they are isomorphic to some $V$, is it true that if $V_1$ and $V_2$ are both submodules/substructures of $V$, then it's an internal direct sum? Is every internal direct sum an external direct sum?

guys would someone help me figure out the differnce between internal and external direct sum?

[direct sum]en.wikipedia.org/wiki/…

Either way, if I understand it right, it should be fine just to take $\langle H,H^a \rangle$, which is subnormal in $H_{(1)}$, and then since $H^b$ is subnormal and satisfies the inductive hypothesis, I can say that their union, which is $\langle H, H^a, H^b \rangle$ is also subnormal in $H_{(1)}$

Or more precisely, I think the claim halfway through about $\langle H, H^a, H^b \rangle$ is wrong - I don't see how it's equal to what it's claimed to be. Also, I think induction should start at s>0, not s>1.

would someone please help me out with the proof starting at page 133 ? The claim is that the derived subgroup of $G$ satisfies ACC (no infinite series of increasing subgroups), then union of two subnormal groups is subnormal.

[title]books.google.cz/…

@TobiasKildetoft Right, thank you Tobias.

Doesn't matter in what sense? That it's easy and it doesn't matter how you do it? (Just pick power set of arr(C), which is still small.? )

Hello, would someone please help me out with a small question - Saunders Mac Lane, Categories book, page 114, proposition 3, why is it said " If the small set J has cardinal larger than arrC" ? Couldn't it be equal to arr C?

Hey guys, has anyone ever heard of "maranda's theorem" in category theory? I'm can't really find what it is, though it does seem it might be this [maranda's completeness theorem]books.google.cz/…

Yeah that's how I understand it.

If we asked the same question for numbers from 10 to 19, including 19, we would have a set {10, ... , 19} (10 elements), and 1 of them would contain a 5 - 15. So there's 10 numbers, 9 don't contain 5, 1 does.

I don't know what you're saying. You have all those numbers, {1000, .... , 8999} and you're asking how many there are of those, that have no 5 in any of their digits.

@TobiasKildetoft Hah that's good advice, I don't understand why I didn't look in some book first...

@MATHASKER I understand it as "a number between 1000 and 9999, such that non of the digits are 5" - the possibilities are - 1st digit 8 (9 possible numbers on the first digit, you can't choose 0), then 9 on 2nd-4th digits. So you have 8*9^3 ?

The universal morphisms one

yes I realize that, but I don't see how it follows from the definition.

@BalarkaSen

I feel like I just shouldn't bother thinking about these questions at this point in time xD

Thanks a lot for the help guys.

But I'm just not sure whether it makes enough sense from a rigorous perspective - it shows there's some sort of thing like a bijection, but I don't know it's a set in the first place, so I'm not sure how it would make sense... it seems like you still need to presume that it's a set. I'm just ranting now.. I'll read the thing you wrote higher up though.

@BalarkaSen Well see that's what I was sort of hoping it says.

I'm not sure what theorem you mean at all.

Were it not directly said it's isomorphic, I'd be fine with just thinking "ok there's some sort of weird bijection between this set/class-thingy and a set"

Well it's said that they're isomorphic, not just in bijection.

@mercio I generally try to avoid these questions given my problems with set theory, and me being unable to tell whether it's worth thinking about too much... but this one time I wanted to understand this a bit rigorously, and I got stuck lol.

@BalarkaSen Sorry, I don't really see how that follows. Would you be willing to please try and fill in a bit more of the reasoning?

Yeah sorry to hear that.

Well if it's possible at all, then that's not so surprising.

Ok, so the elements of that set themselves aren't sets, but it's a still a set hm. I didn't even know that's possible.

@BalarkaSen If it's locally small, but the domain category's objects arent' a set, how can I tell that a specific natural transformations between the two functors is a set?

That would make sense, but I don't see that the domain category is small - there's written it's locally small.