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Math Help - Isomorphism between groups...

  1. #1
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    Isomorphism between groups...

    Hello!

    I am trying to show that \alpha is an ismorphism,
    \alpha:U(st) \rightarrow U(s) \oplus U(t)
    defined by \alpha(x) = (x\ mod\ s, x\ mod\ t) where x \in U(st)
    where U(n) is the multiplicative group of units (refresher: numbers relativly prime to n)
    and \oplus is the external direct product.

    so clearly \alpha is well defined
    I have shown it is 1-1
    So my problem is...
    I am having trouble showing is it onto (to show to that it is a bijection)
    so let b = (y_1,y_2) \in U(s) \oplus U(t). So b is an arbitrary element of U(s) \oplus U(t).

    so what we need to do is find some x \in U(st) such that
    \alpha(x) = b = (y_1,y_2) (we must find a x that maps to b... defn of onto, right?)

    Keep in mind that we know that gcd(s, y_1) = gcd (t, y_2) = 1 since gcd(s,t) = 1

    --------
    So I thought maybe x = y_1t +y_2s?

    then by the definition of alpha,

    \alpha(x) = ((y_1t +y_2s)\ mod\ s, (y_1t +y_2s)\ mod\ t) = ((y_1t)\ mod \ s, (y_2s)\ mod\ t)
    Now we just need to get rid of the t in the first component and s in the second component to map to that arbitrary element b = (y_1, y_2) \in U(s) \oplus U(t)

    Yikes sorry for all the description. All help would be much appriciated, this actually stumped our class for a few minutes at the end of class today!


    Thanks! in advance!
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  2. #2
    Senior Member roninpro's Avatar
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    You can cop out a little bit by noting that your map is between finite sets. Since the map is injective, it is automatically surjective.
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  3. #3
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by roninpro View Post
    You can cop out a little bit by noting that your map is between finite sets. Since the map is injective, it is automatically surjective.
    He'd also have to note that the sets are of the same size!
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  4. #4
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    which they are!
    \phi(mn) = \phi(m)\phi(n) for Euler's phi function (which counts the number of elements in U(n).

    thanks alot! I had totally forgotten that handy tool.

    just for interests sake, still wouldn't mind to find an element mapping to b though!
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  5. #5
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by matt.qmar View Post
    which they are!
    \phi(mn) = \phi(m)\phi(n) for Euler's phi function (which counts the number of elements in U(n).

    thanks alot! I had totally forgotten that handy tool.

    just for interests sake, still wouldn't mind to find an element mapping to b though!
    Don't you need to assume that (s,t)=1?
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  6. #6
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    Yes, only holds for coprimes. But in this case, gcd(s,t)=1
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