Let $n > 1$ be an integer and let $\theta = \dfrac{2\pi}{n}$. Let $P$ be the regular $n$-gon with vertices ($\cos i\theta$, $\sin i \theta$) for $i \in \mathbb Z_n$. The dihedral group $D_n$ is the symmetry group of $P$, which consists of rotations $R_i$ and reflections $F_i$ for $i \in \mathbb Z_n$.
$R_i$ is the counterclockwise rotation around the origin by angle $i$, and $F_i$ is the reflection across the line through the origin by angle $i\theta$ and $F_i$ is the reflection across ($\cos i\theta_2$, $\sin i\theta_2$).
How to give general formulas for $R_iR_j$ , $R_iF_j$ , $F_iR_j$ , and $F_iF_j$ . For example, $R_iR_j = R_{i+j}$ ,where the addition of indices is $\mod n$
And is $D_n$ a group? If not, what is missing?