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Dihedral group - isomorphism

  1. Aug 26, 2011 #1
    The dihedral group Dn of order 2n has a subgroup of rotations of order n and a subgroup of order 2. Explain why Dn cannot be isomorphic to the external direct product of two such groups.

    Please suggest how to go about it.

    If H denotes the subgroup of rotations and G denotes the subgroup of order 2.

    G = { identity, any reflection} ( because order of any reflection is 2)

    I can see that order of Dn= 2n = order of external direct product
     
  2. jcsd
  3. Aug 26, 2011 #2

    micromass

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    Try to show that Dn is nonabelian (for [itex]n\geq 3[/itex]) and that your direct product will always be abelian...
     
  4. Aug 31, 2011 #3
    Thanks a lot. I got it.

    If we take H as the subgroup consisting of all rotations of Dn, then being a cyclic group, it would also be abelian. Then again, subgroup K of order 2 is abelian.

    Further, the external direct product H + K is abelian as H and K are abelian.

    Thanks !!
     
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