# Thread: Help with Quotient Groups

1. ## Help with Quotient Groups

So I really feel lost on Quotient groups, I just dont get the concept. My book gives a simple example...

( $\mathbb{Z} \oplus \mathbb{Z}$)/ ( $\mathbb{Z} \oplus$ <0>)

Could someone explain to me what the construction of this is explicitly and what its structure is?

Im really hoping that I can get the concept with this example. Thanks

2. Sart from the coset!
can you find all the cosets ? Do you see that it is similar to something we have learned for years?
Do not be confused by the notation, try to understand what the representation really is.

But the truth is I just dont see it. I dont really know what the cosets look like.

Thats why I asked to demonstrate it as an example.

4. Originally Posted by ElieWiesel

But the truth is I just dont see it. I dont really know what the cosets look like.

Thats why I asked to demonstrate it as an example.

Well, then try to find a homom. from $\mathbb{Z}\oplus\mathbb{Z}$ to some rather well-known group and s.t. the subgroup $\mathbb{Z}\oplus{0}$ is the kernel...

Tonio

5. Look, I appreciate you taking the time to write things.

But I dont understand the idea of a quotient group from the very beginning.

The reason I used this example, because It is a problem out of the book. I thought it was complex enough, yet simple enough to demonstrate the idea.
I need you to tell what im looking at. I need you to tell me what it is i should see when I see this type of notation.

Please just write what it is. So I can learn from it.

6. Originally Posted by ElieWiesel
Look, I appreciate you taking the time to write things.

But I dont understand the idea of a quotient group from the very beginning.

The reason I used this example, because It is a problem out of the book. I thought it was complex enough, yet simple enough to demonstrate the idea.
I need you to tell what im looking at. I need you to tell me what it is i should see when I see this type of notation.

Please just write what it is. So I can learn from it.
Have you covered the 1st Isomorphism Theorem yet? This is a very nice theorem which basically says quotients and homomorphic images are the same thing. Homomorphic images are, in some respects, quite easy to understand, while quotients are not. So, I shall explain what a homomorphic image is, and then you can apply this to quotients.

Basically, a homomorphic image of a group is another group which preserves all the rules of your previous group (if the pre-image is abelian, so will the image, if $ab=c$ then $(a \theta)(b\theta)=c\theta$, etc.) However, other we can add other requirements to the group. For instance, if we take a non-abelian group we could force the image to be abelian (formally, we quotient out something called the derived subgroup). If you think about the kernel of this image, $aba^{-1}b^{-1}$ will be in this kernel. So, we have added a restriction to the image - the image is abelian. We can do lots of things - we can, for instance, dictate that every element has finite order.

However, there are some things you cannot do. Everything must conform to the operations of the group in the pre-image. For instance, if you have an element of order $a$ in the pre-image then it must have an order that divides $a$ in the image.

Basically, homomorphic images, and thus quotients, restrict the group we are looking at. At least, that is how I look at them.

I hope I haven't confused you too much.