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Abstract Algebra dihedral group

  1. Sep 27, 2011 #1
    1. The problem statement, all variables and given/known data
    Let G be a finite group and let x and y be distinct elements of order 2 in G that generate G. Prove that G~=D_2n, where |xy|=n.

    I have no idea how to solve this or even where to begin. I tried setting up G=<x,y|x^2=y^2=1=(xy)^n> But couldn't get any farther, I am so close to dropping this class please help!
     
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  3. Sep 27, 2011 #2

    micromass

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    So, what do you all know about the dihedral group?? How is it defined?? What are its characterizations?
     
  4. Sep 27, 2011 #3
    I know that the D_2n = <r,s|r^9=s^2=1, rs=sr^-1> and that is our dihedral group.
     
  5. Sep 27, 2011 #4

    micromass

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    You probably mean

    [tex] D_{2n} = <r,s|r^n=s^2=1, rs=sr^{-1}> [/tex]

    Now, put r=xy and s=y. Show that the relations hold.
     
  6. Sep 27, 2011 #5
    oops yes i wrote 9 instead of n and i did that I showed that its surjective and that its a homomorphism. However, I am stuck on how to make it show that its injective...
     
  7. Sep 27, 2011 #6

    micromass

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    What is an homomorphism?? I'm not following you...
     
  8. Sep 27, 2011 #7
    well we want to show that G and D_2n is isometric and in order to do that we show its surjective, injective and a homomorphism
     
  9. Sep 27, 2011 #8

    micromass

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    Show that what is surjective, injective and homomorphism???? You didn't give a map yet. What is the map between G and [itex]D_{2n}[/itex]????
     
  10. Sep 27, 2011 #9
    our map is phi:D_2n->G where phi(r)=(xy), and phi(s)=x
     
  11. Sep 27, 2011 #10

    micromass

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    Can you think of an inverse mapping??
     
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