Proposition 7

The group W is dihedral (Chap. IV, § 1, no. 2) *of order $2m$. It consists of the rotations $\rho_{2n\theta}$ and the products $\rho_{2n\theta} s$ for $n \in \mathbf{Z}$. The transforms of D and D' by the elements of W are the transforms of D by the rotations $\rho_{n\theta}$ for $n \in \mathbf{Z}$.

The Corollary to Prop. 6 shows that $s's$ is of order m, which gives the first assertion. The elements of W are thus of the form $(s's)^n = \rho_{2n\theta}$ or $(s's)^n s = \rho_{2n\theta} s$; the last assertion follows from this, since $D' = \rho_\theta(D)$.