Abstract Algebra dihedral group

[corey2014]

Homework Statement

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!

 

[micromass]

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

 

[corey2014]

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

 

[micromass]

I know that the D_2n = <r,s|r^9=s^2=1, rs=sr^-1

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.

 

[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...

 

[micromass]

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

 

[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

 

[micromass]

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

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

 

[corey2014]

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

 

[micromass]

Can you think of an inverse mapping??