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Math Help - Group and subgroup

  1. #1
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    Group and subgroup

    (G;x) is a group ( e it's identity element). H_1 and H_2 are a subgroups from (G,x).
    H_1H_2={ x_1x_2/ (x_1;x_2} \in H_1x H_2}

    1)-I must show that : H_1H_2 is a subgroup from(G;x) \Longrightarrow H_1H_2=H_2H_1
    2)- We assume that H_1 and H_2 are finished and H_1\cap H_2={e} and:
    \phi : H_1x H_2 \longrightarrow H_1 H_2
    ( x_1; x_2) \longrightarrow x_1 x_2

    I must show that \phi is surjective and Card( H_1H_2)=Card( H_1)Card( H_2)
    I don't know anything!!!!!!! Can you give me some help please????
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bhitroofen01 View Post
    Let H_1,H_2\leqslant G be such that H_1\cap H_2=\{e\}. Define \phi:H_1\times H_2\to H_1H_2 by (h_1,h_2)\overset{\phi}{\longmapsto}h_1h_2. Prove that \phi is surjective.
    Is the above the second part?
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  3. #3
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    yes
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by bhitroofen01 View Post
    yes
    Let's see some work.
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  5. #5
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    H_1 H_2 is a subgroup from (G;x) \Longleftrightarrow H_1x( H_2)^{-1}\in H_1H_2
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  6. #6
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    Did you drop any assumptions? The statement doesn't seem true to me.
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