For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.
[(a)] Verify that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, there exists a unique $g^{\prime\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{g^{\prime\prime}} = \phi_{g^{\prime}} \circ \phi_{g}$. We denote this element by $g^{\prime}g$ in the following.
[(b)] Verify that for all $g_{1}, g_{2}$ and $g_{3}$ in $\operatorname{Dep}(\mathbb{R}^{d})$ we have $g_{1}(g_{2}g_{3}) = (g_{1}g_{2})g_{3}$.
For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.
\begin{itemize}
\item[(a)] Verify that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, there exists a unique $g^{\prime\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{g^{\prime\prime}} = \phi_{g^{\prime}} \circ \phi_{g}$. We denote this element by $g^{\prime}g$ in the following.
\item[(b)] Verify that for all $g_{1}, g_{2}$ and $g_{3}$ in $\operatorname{Dep}(\mathbb{R}^{d})$ we have $g_{1}(g_{2}g_{3}) = (g_{1}g_{2})g_{3}$.
\end{itemize}