Uniquely n-Divisible and Torsionfree Groups in Exact Sequences

[mathsss2]

Let $$n$$ be a nonzero integer. An abelian group $$A$$ is called $$n$$-divisible if for every $$x \in A$$, there exists $$y \in A$$ such that $$x=ny$$. An abelian group $$A$$ is called $$n$$-torsionfree if $$nx=0$$ for some $$x \in A$$ implies $$x=0$$. An abelian group $$A$$ is called uniquely $$n$$-divisible if for any $$x \in A$$, there exists exactly one $$y \in A$$ such that $$x=ny$$.
Let $$\mu_n : A \rightarrow A$$ be the map $$\mu_n(a)=na$$
(a) Prove that $$A$$ is $$n$$-torsionfree iff $$\mu_n$$ is injective and that $$A$$ is $$n$$-divisible iff $$\mu_n$$ is surjective.
(b) Now suppose $$0 \rightarrow A \overset{f}{\rightarrow} B \overset{g}{\rightarrow} C \rightarrow 0$$ is an exact sequence of abelian groups. It is easy to check that the following diagram commutes: [see attachment]
Suppose that $$B$$ is uniquely $$n$$-divisible. Prove that $$C$$ is $$n$$-torsionfree if and only if $$A$$ is $$n$$-divisible.

 

[Hurkyl]

It's hard to be helpful if I have no idea where you're stuck...

 

[mathsss2]

So I am now thinking to use the snake lemma for part b). However, I am a little confused on how to do this. Also, for part a) these statements seem a bit trivial, I will work on those proofs using the definitions. Do you have any ideas on part a) or b), I never used the snake lemma to prove something yet.

 

[Hurkyl]

I agree, (a) looks easy. I think it's just a matter of writing out the definitions, as you say.

For (b) you have the right idea -- in the two cases (assuming C n-torsion-free and assuming A n-divisible), what did you compute as the sequence you get from the snake lemma?

 

[mathsss2]

So here is what I have for the first part of a).

$$A$$ is $$n$$-torsionfree
$$\Leftrightarrow \forall x\in A\ (nx=0 \Rightarrow x=0)$$
$$\Leftrightarrow \forall x\in A\ (\mu_n(x)=0 \Rightarrow x=0)$$
$$\Leftrightarrow \mu_n$$ is injective

I am confused now on how to do the second part of a).

 

[mathsss2]

How do I do the second part with the snake lemma? I do not know how to do this part.

Here is the picture:
http://i719.photobucket.com/albums/ww191/xianghu21/2.png?t=1229062259"

 

[Hurkyl]

So here is what I have for the first part of a).

$$A$$ is $$n$$-torsionfree
$$\Leftrightarrow \forall x\in A\ (nx=0 \Rightarrow x=0)$$
$$\Leftrightarrow \forall x\in A\ (\mu_n(x)=0 \Rightarrow x=0)$$
$$\Leftrightarrow \mu_n$$ is injective

I am confused now on how to do the second part of a).

Doesn't the same idea work? What went wrong?

How do I do the second part with the snake lemma? I do not know how to do this part.

Here is the picture:
http://i719.photobucket.com/albums/ww191/xianghu21/2.png?t=1229062259"

Can you be more precise? Is it... that you don't know how to write down the exact sequence given by the snake lemma? Is it... that you don't know what to do with the sequence one you have it? Is it... something else?

 

[mathsss2]

So, I need to take the kernels and cokernels of the vertical mappings and apply the snake lemma. Also, I need to write down the exact sequence to use the snake lemma. I am not sure how to actually do this. I never used the snake lemma to prove something before. I guess I just need to know how to start.

 

[mathsss2]

Here is how to do the second part of part a):

$$\mu_n \text{ is surjective}$$
$$\Leftrightarrow \forall y\in A,\, \exists x\in A \, (\mu (x) = y)$$
$$\Leftrightarrow \forall y\in A,\, \exists x\in A\, (nx = y)$$
$$\Leftrightarrow \text{ A is n-divisible}$$

For part b):
I am still not sure how to do part b).

 

[Hurkyl]

So, I need to take the kernels and cokernels of the vertical mappings and apply the snake lemma. Also, I need to write down the exact sequence to use the snake lemma. I am not sure how to actually do this.

I don't see what the problem is. I presume you know what the 6 groups are -- you even have concrete descriptions of many of them. (As opposed to a purely formal description, such as "$\\ker \left( B \xrightarrow{\mu_n} B \right)$") I presume you even know how those groups are to be organized into an exact sequence. So I don't understand why you are not sure how to write the exact sequence.

 

[mathsss2]

I still do not know how to solve (b) using the snake lemma or any other ways. help!
When I use the snake lemma, how will this help me with what I am trying to show? And for both directions? I am very confused.

just solved it. :-)

 

[Hurkyl]

When I use the snake lemma, how will this help me with what I am trying to show?

From the information you've given me, it sounds like you haven't even bothered to write down what the snake lemma tells you in this case! And if you haven't even done that, then of course you don't know how it will help!

It's okay to try something without knowing how it will be useful. In fact, doing so is required for doing mathematics. (In fact, it's a requirement for doing research in any field!)

So... just do it! Write down what the snake lemma tells you, and figure out everything you can from there. If you can't manage to see your way to the end of the problem, then come back here and show us how far you have gotten, and we can help you figure out what you've overlooked.

 

[mathsss2]

I solved it Hurkyl (as I said in my post if you read it thoroughly). And no, I didn't use the snake lemma at all. One step was messing me up, but I got through it.

 

[adriank]

To me it looks like an easy application of the four lemmas, no? Since

• A is n-torsion free ⇔ μ_n is a monomorphism
• A is n-divisible ⇔ μ_n is an epimorphism
• A is uniquely n-divisible ⇔ μ_n is an isomorphism

Moreover, it proves the stronger statement that if B is n-divisible and C is n-torsion free then A is n-divisible, and if B is n-torsion free and A is n-divisible, then C is n-torsion free. I'm pretty sure that was the point of part (a) actually: setting it up for proving (b).