Symplectic group

The only scalars in ${\mathrm{Sp}}_{2m}(q)$ are $\left\{\pm I_{2m}\right\}$. Hence

$$\mathrm{PSp}(V):=\mathrm{Sp}(V)/\left\{\pm I_{2m}\right\}=\mathrm{Sp}(V)/\mathrm{Z}(\mathrm{Sp}(V)),$$

where $\mathrm{Z}(\mathrm{Sp}(V))=\left\{\pm I_{2m}\right\}$.

Theorem

The group ${\mathrm{PSp}}_{2m⩾4}(q)$ is simple, except for ${\mathrm{PSp}}_4(2)\cong S_6$.