  We give some properties of the subgroup $G_n(\mathbb{C})$ of the group of
birational self-maps of $\mathbb{P}^n_\mathbb{C}$ generated by the standard
involution and the group of automorphisms of $\mathbb{P}^n_\mathbb{C}$. We
prove that there is no nontrivial finite-dimensional linear representation of
$G_n(\mathbb{C})$. We also establish that $G_n(\mathbb{C})$ is perfect, and
that $G_n(\mathbb{C})$ equipped with the Zariski topology is simple.
Furthermore if $\varphi$ is an automorphism of
$\mathrm{Bir}(\mathbb{P}^n_\mathbb{C})$, then up to birational conjugacy, and
up to the action of a field automorphism $\varphi_{\vert G_n(\mathbb{C})}$ is
trivial.
