Show that $\alpha$ is a group morphism and describe its kernel, where $$\begin{aligned} \alpha : S \times S & \longrightarrow \mathrm{GL}(\mathbb{H}) \\ (u, v) & \longmapsto (Z \mapsto uZv^{-1}) \end{aligned}$$
Show that $\alpha$ is a group morphism and describe its kernel, where
$$\begin{aligned} \alpha : S \times S & \longrightarrow \mathrm{GL}(\mathbb{H}) \\ (u, v) & \longmapsto (Z \mapsto uZv^{-1}) \end{aligned}$$