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Math Help - Fun with normal subgroups.

  1. #1
    Junior Member
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    Fun with normal subgroups.

    This stuff drives me bonkers. Anyway, I have been given or have proved the following:

    \phi : X \rightarrow Y is onto.

    x, \bar{x} \in X

    U \subset X is a subgroup

    \phi (U) is a subgroup of Y

    ---------------------

    My last question (for this section) is:

    Verify that if U is normal in X, then \phi (U) is normal in Y

    -------------

    Can I say this:

    If U is normal in X then:

    uXu^{-1}=X for all u \in U , so:

     \phi(uXu^{-1})= \phi (X)

    \phi(u)\phi(X) \phi(u^{-1})= \phi (X)

    \phi(u) Y \phi(u)^{-1}= Y for all \phi(u) \in \phi(U)

    Hence, \phi(U) is normal in Y
    Last edited by MichaelMath; October 25th 2010 at 10:59 AM.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    That's perfect. Of course, it all comes down to the fact that \phi(u) can be any element of Y. You should perhaps write that down explicitly.
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