# Math Help - Generators and relations

1. ## Generators and relations

Use generators and relations to show that if $x\in\text{D}_{2n}$ which is not a power of r, then $rx=xr^{-1}$

$\text{D}_{2n}=$

Basically, x in this case is a flip. I am not sure what needs to proved. x is just a flip so by definition it satisfy $rx=xr^{-1}$

2. ## Re: Generators and relations

since x is not a power of r it will be a multiplication of power of r and s say

$x = r^k s$

we have to show that $r r^k s = r^k s r^{-1}$

3. ## Re: Generators and relations

Originally Posted by Amer
since x is not a power of r it will be a multiplication of power of r and s say

$x = r^k s$

we have to show that $r r^k s = r^k s r^{-1}$
Can I do this:

Since $rs=sr^{-1}$, $r^ksr^{-1}=r^krs=r^{k+1}s\equiv rr^ks=r^{k+1}s$

4. ## Re: Generators and relations

Originally Posted by dwsmith
Can I do this:

Since $rs=sr^{-1}$, $r^ksr^{-1}=r^krs=r^{k+1}s\equiv rr^ks=r^{k+1}s$
Yup, that works.