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Math Help - Help with Quotient Groups

  1. #1
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    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
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  2. #2
    Senior Member Shanks's Avatar
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    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.
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  3. #3
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    Thanks for your help.

    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.

    Thanks in advance
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  4. #4
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    Quote Originally Posted by ElieWiesel View Post
    Thanks for your help.

    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.

    Thanks in advance

    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
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  5. #5
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    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.
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  6. #6
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by ElieWiesel View Post
    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.
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