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Math Help - D4 question

  1. #1
    nhk
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    D4 question

    Prove that D4 (Dihedral group) cannot be expressed as an internal direct product of two proper subgroups.
    I know that the only two possible subgroups would be the subgroups of order 4 and 2. I am thinking since D4 is not commutative I can get a contradicition this way, but I am not sure how to do it. Any help would be welcomed
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nhk View Post
    Prove that D4 (Dihedral group) cannot be expressed as an internal direct product of two proper subgroups.
    I know that the only two possible subgroups would be the subgroups of order 4 and 2. I am thinking since D4 is not commutative I can get a contradicition this way, but I am not sure how to do it. Any help would be welcomed
    I thought the dihedral group D_n is always isomorphic to a semidirect product of C_2 and C_n
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  3. #3
    nhk
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    what is C2 and Cn? I think we have not got that advanced in the class i am taking right now.
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  4. #4
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nhk View Post
    what is C2 and Cn? I think we have not got that advanced in the class i am taking right now.
    What are you doing in class right now?
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  5. #5
    nhk
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    factor groups and normal subgroups are what we are doing in my class right now.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by nhk View Post
    factor groups and normal subgroups are what we are doing in my class right now.
    I'm not sure. You should probably let someone better at group theory (e.g. tonio) take a look at this. I'm pretty sure that D_4\cong C_4\ltimes  C_2
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  7. #7
    nhk
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    tnahks for your help anyway.
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  8. #8
    nhk
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    tonio can you save me again please?
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    Quote Originally Posted by nhk View Post
    tonio can you save me again please?
    Well, I might be able to help u too

    As u said, the possible proper subgroups of D4 are all abelian (groups of order 2, 4). If D4 are the internal direct product of its nontrivial proper subgroups, then D4 becomes abelian. Contradiction.

    As Drexel28 said,

    D_4 \cong C_4 \ltimes_\phi C_2, where \phi:C_2 \rightarrow \text{Aut}(C_4) and the assoicated action is x \cdot h = h^{-1} for all h in C_4 and x in C_2 such that xhx^{-1} = h^{-1}.
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  10. #10
    nhk
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    Quote Originally Posted by aliceinwonderland View Post
    Well, I might be able to help u too

    As u said, the possible proper subgroups of D4 are all abelian (groups of order 2, 4). If D4 are the internal direct product of its nontrivial proper subgroups, then D4 becomes abelian. Contradiction.

    As Drexel28 said,

    D_4 \cong C_4 \ltimes_\phi C_2, where \phi:C_2 \rightarrow \text{Aut}(C_4) and the assoicated action is x \cdot h = h^{-1} for all h in C_4 and x in C_2 such that xhx^{-1} = h^{-1}.
    I am not sure why that the internal direct product of abelain subroups makes D4 abelian, could you explain this to me a little?
    Thanks for your help
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  11. #11
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    Quote Originally Posted by nhk View Post
    I am not sure why that the internal direct product of abelain subroups makes D4 abelian, could you explain this to me a little?
    Thanks for your help
    Direct sum - Wikipedia, the free encyclopedia
    Direct sum - Wikipedia, the free encyclopedia
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  12. #12
    nhk
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    definitions always get me!! THanks for you help aliceinwonderland
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