 Symplectic group over a finite field

This group preserves a nondegenerate alternating bilinear form on ${\mathbb{F}}_q^{2m}$. Choosing a nonzero vector and a partner of pairing one, then recursing on their orthogonal complement, gives $|S{p}_{2m}(q)|=q^{m^2}{\prod }_{i=1}^m(q^{2i}-1)$. Its center consists of scalars with ${\lambda }^2=1$.