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Math Help - Isomorphism (Help)

  1. #1
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    Isomorphism (Help)

    If H and K are normal subgroups of a group G with HK=G, prove that G/(H \cap K) \simeq (G/H) \times (G/K).
    Hint: If \phi:G \rightarrow (G/H) \times (G/K) is defined by x \mapsto (xH,xK), then ker \: \phi=H \cap K; moreover, we have G=HK, so that \displaystyle\cap_{a}aH=HK=\displaystyle\cap_{b}bK.

    I have managed to prove that ker \: \phi=H \cap K. I'm having difficulty in proving that \phi is surjective. This is crucial in proving the above isomorphism using 1st Isomorphism Theorem.
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Markeur View Post
    If H and K are normal subgroups of a group G with HK=G, prove that G/(H \cap K) \simeq (G/H) \times (G/K).
    Hint: If \phi:G \rightarrow (G/H) \times (G/K) is defined by x \mapsto (xH,xK), then ker \: \phi=H \cap K; moreover, we have G=HK, so that \displaystyle\cap_{a}aH=HK=\displaystyle\cap_{b}bK.

    I have managed to prove that ker \: \phi=H \cap K. I'm having difficulty in proving that \phi is surjective. This is crucial in proving the above isomorphism using 1st Isomorphism Theorem.
    Let \left(yH,zK\right) be in the codomain. Since HK=G what can you say about y,z?
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  3. #3
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    Hey,

    The proof is as below:

    We want to show that \forall (yH,zK) \in (G/H) \times (G/K), \exists g \in G such that \phi(g)=(gH,gK).

    Since (yH,zK) \in (G/H) \times (G/K), yH \in G/H and zK \in G/K.
    Since G=HK, yH \in HK/H and zK \in HK/K.
    This implies that y,z \in HK.
    Since G=HK, g \in HK.
    This implies that y=g and z=g \: \forall y,z \in G.
    Hence \exists g \in G such that \phi(g)=(gH,gK).
    Therefore (G/H) \times (G/K) \subseteq Im \; \phi.
    Clearly, Im \; \phi \subseteq (G/H) \times (G/K).
    Finally, we conclude that (G/H) \times (G/K)=Im \; \phi.

    From there, we can establish the isomorphism using First Isomorphism Theorem.

    So, is the above proof correct?

    Thanks in advance.
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