Example

Let $M$ be the multiplicative group $\{1, -1\}$, and let $m$ be an integer $\geq 2$ (resp. $m = \infty$). Then $M$ acts on the group $\mathbf{Z}/m\mathbf{Z}$ (resp. on $\mathbf{Z}$) by $(-1).x = -x$, and the corresponding semi-direct product of $M$ by $\mathbf{Z}/m\mathbf{Z}$ (resp. of $M$ by $\mathbf{Z}$) is denoted by $D_m$. The elements of $D_m$ are thus the pairs $(\varepsilon, x)$ with $\varepsilon = \pm 1$ and $x \in \mathbf{Z}/m\mathbf{Z}$ (resp. $x \in \mathbf{Z}$); the group law on $D_m$ is given by the formula

We denote by $\iota$ the class of 1 modulo $m$ (resp. $\iota = 1$) and set Then $\rho^2 = {\rho'}^2 = 1$ and $\pi = \rho \rho'$. The formulas show that $D_m$ is a dihedral group generated by $\{\rho, \rho'\}$.