Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 2 / 6 / a / Solution

Let $V$ have dimension $2m$ over the finite field ${\mathbb{F}}_q$, with a nondegenerate alternating bilinear form $B$. The symplectic group over a finite field isIt acts simply transitively on ordered symplectic bases: a linear map between two such bases is uniquely specified and preserves all pairings, hence the whole bilinear form.

Count a first hyperbolic pair $(e_1,f_1)$. There are $q^{2m}-1$ choices for $e_1\ne 0$. Nondegeneracy makes $B(e_1,-)$ a nonzero linear functional, so there are $q^{2m-1}$ choices of $f_1$ with $B(e_1,f_1)=1$. Their span is nondegenerate, and its symplectic orthogonal complement is a symplectic vector space of dimension $2m-2$. Recursing, with $|S{p}_0(q)|=1$, yieldsThe construction also proves the existence of the required symplectic bases, in characteristic two as well as odd characteristic.