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DIHEDRAL GROUP - Internal Direct Product

  1. Sep 23, 2011 #1
    I have to prove that D4 cannot be the internal direct product of two of its proper subgroups.Please help.
     
    Last edited: Sep 23, 2011
  2. jcsd
  3. Sep 23, 2011 #2

    micromass

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    So, what did you try already??
    What can the orders be of a proper subgroup of D4?? Can they be abelian, nonabelian?
     
  4. Sep 23, 2011 #3

    Thanks, Micromass. If G is the internal direct product of its subgroups H and K ,then the possible orders of subgroups H and K can be 2 and 4 or vice a versa.

    It seems that both H and K are abelian. How can move further from this ?
     
  5. Sep 23, 2011 #4

    micromass

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    Indeed, an the direct product of abelian groups is...
     
  6. Sep 23, 2011 #5
    As far as the result goes the external direct product of two abelian groups is abelian....but is the internal direct product abelian too ? i mean if subgroups H and K are abelian can we conclude that the IDP is abelian ?
     
  7. Sep 23, 2011 #6

    micromass

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    Well, the internal direct product of H and G is isomorphic to the external direct product if [itex]H\cap G=\{e\}[/itex]. Use that.
     
  8. Sep 25, 2011 #7
    Thanks a lot, Micromass.
     
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