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Math Help - isomorphic group

  1. #1
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    isomorphic group

    Is 1Z/14Z isomorphic to 2Z/28Z? How do I solve or prove this?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by grace412 View Post
    Is 1Z/14Z isomorphic to 2Z/28Z? How do I solve or prove this?
    Yes. Think about the order of your groups. If x is the generator of Z, then in Z/14Z it(s image) has order 14 (obviously), so what is the order of (the image of) 2x in 2Z/28Z? Both groups are cyclic, so...

    (Another way you can think about this is index: What is the index of 28Z in 2Z? and 14Z in Z? As images of cyclic groups are cyclic then...)
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    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by grace412 View Post
    Is 1Z/14Z isomorphic to 2Z/28Z? How do I solve or prove this?
    define a map
    defined as:


    show now that is homomorphism, injection and surjection.
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  4. #4
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    if you follow the advice of the post above, be sure to also show that \phi is well-defined.
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