$D_n = \langle \mu, \rho \mid \mu^2 = \rho^n = \epsilon, \mu\rho = \rho^{-1}\mu \rangle$
Let $n=dm$ and consider the group $D_n$, with $K = \langle \rho^m \rangle$. Show that the following $2m$ cosets of $K$ are all distinct: $K, \rho K, \rho^2 K, \dots , \rho^{m-1}K, \mu K, \mu \rho K, \mu \rho^2 K, \dots, \mu \rho^{m-1}K$.
My Approach:
First, I need to prove that all cosets of the form $\rho K$ are distinct then I'll show all cosets of the form $\mu \rho^x K$ are distinct then I'll compare both of these forms of cosets and show they are distinct. Assume that $\rho^xK =\rho^yK$ where $x$ and $y$ are distinct and less than $m$. this imples that $\rho^{-x}*\rho^y \in K$. However, whatever I do I can't find a contradiction.