We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$. Let $H$ be the set of matrices $M(A, \vec{b})$ of $G$ such that $\Psi(M(A, \vec{b})) = \Delta\left(0, \vec{e}_1\right)$. a) Describe the elements of $H$. b) Show that $H$ is a subgroup of $G$. c) Show that for all $g$ in $G$ and all $h$ in $H$, we have $\Psi(gh) = \Psi(g)$.
We denote by $\mathcal{D}$ the set of affine lines of the plane and we consider the application $\Psi : \left\{ \begin{array}{cll} G & \rightarrow & \mathcal{D} \\ M(A, \vec{b}) & \mapsto \Delta\left(\left\langle A\vec{e}_1, \vec{b}\right\rangle, A\vec{e}_1\right) \end{array} \right.$.
Let $H$ be the set of matrices $M(A, \vec{b})$ of $G$ such that $\Psi(M(A, \vec{b})) = \Delta\left(0, \vec{e}_1\right)$.
a) Describe the elements of $H$.
b) Show that $H$ is a subgroup of $G$.
c) Show that for all $g$ in $G$ and all $h$ in $H$, we have $\Psi(gh) = \Psi(g)$.