Symplectic group over a finite field
= Symplectic group over a finite field
{title2=$Sp_{2m}(q)=\{g:g^{\mathsf T}Jg=J\}$}
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 $|Sp_{2m}(q)|=q^{m^2}\prod_{i=1}^m(q^{2i}-1)$. Its center consists of scalars with $\lambda^2=1$.