Suppose we have a group $A$ which is generated by generators $R$ and $F$, subject to the relation $$ R^n=I, F^2=I,RF= FR^{-1}.$$
It should be just the dihedral group of order $2n$, the one generated by $n$ rotations and 1 reflection.
I am a bit confused that how I show these relation will define $A$ to have exactly $2n$ elements.
I understand that there would be $I, R,\dots, R^{n-1}$ and $F,RF,\dots, R^{n-1}F$. But how can I show these elements are distinct. For example, how can I show $R\not= R^2F$. The notes I learned just says that if they are equal then we would have $F=R^4$ and it is not possible to get this using the relations. I am a bit worried here. How do you see that these relation does not imply $F=R^4$ or more generally $F= R^{k}$ for any $k$.
I guess this should be really basic but I don't see how.