Abstract Algebra dihedral group

1. Sep 27, 2011

corey2014

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!

2. Sep 27, 2011

micromass

Staff Emeritus
So, what do you all know about the dihedral group?? How is it defined?? What are its characterizations?

3. Sep 27, 2011

corey2014

I know that the D_2n = <r,s|r^9=s^2=1, rs=sr^-1> and that is our dihedral group.

4. Sep 27, 2011

micromass

Staff Emeritus
You probably mean

$$D_{2n} = <r,s|r^n=s^2=1, rs=sr^{-1}>$$

Now, put r=xy and s=y. Show that the relations hold.

5. Sep 27, 2011

corey2014

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...

6. Sep 27, 2011

micromass

Staff Emeritus
What is an homomorphism?? I'm not following you...

7. Sep 27, 2011

corey2014

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

8. Sep 27, 2011

micromass

Staff Emeritus
Show that what is surjective, injective and homomorphism???? You didn't give a map yet. What is the map between G and $D_{2n}$????

9. Sep 27, 2011

corey2014

our map is phi:D_2n->G where phi(r)=(xy), and phi(s)=x

10. Sep 27, 2011

micromass

Staff Emeritus
Can you think of an inverse mapping??