Example 1

Let m be an integer $\geq 2$ or $\infty$ and let W be a group defined by a set of generators S = {s, s'} and the relations $s^2 = {s'}^2 = 1$ when $m = \infty$, $s^2 = {s'}^2 = (ss')^m = 1$ when m is finite. Consider on the other hand the dihedral group D_m (no. 2, Example) and the elements $\rho$ and $\rho'$ of D_m defined by (7). Since $\rho^2 = (\rho')^2 = 1$ and $(\rho\rho')^m = 1$ when m is finite, there exists a unique homomorphism f from W to D_m such that $f(s) = \rho$ and $f(s') = \rho'$. Since $\rho\rho'$ is of order m, it follows that ss' is itself of order m. Hence, (W, S) is a Coxeter system, W is a dihedral group of order $2m$ and f is an isomorphism (Prop. 2).

By transport of structure, it follows that every dihedral group is a Coxeter group.