Order formulas for symplectic groups

This article describes order formulas for the symplectic group of finite degree over a finite field and its variants.

For a finite field of size $q$

Explanation for order of general linear group

We describe here the reasoning behind the formula for the order of the general linear group $Sp(2m,q)=Sp(2m,\mathbb {F} _{q})$.

The order equals the number of choices of basis for $\mathbb {F} _{q}^{2m}$ where the basis is an ordered symplectic basis: the basis comes in the form of an ordered collection of $m$ ordered pairs. The conditions are as follows: with respect to the original symplectic form, the two-dimensional subspaces spanned by these are mutually orthogonal, and the form applied to the first and second vector within each pair gives the value 1.

The order of the symplectic group is the number of possible pairs of this sort.

For the first vector (say $v_{1}$) of the first pair, there are $q^{2m}-1$ choices. Let's say $w_{1}$ is the second vector of the first pair. The bilinear form gives a linear functional $x\mapsto b(v_{1},x)$ which is nonzero. We want $w_{1}$ to be in the pre-image of 1 under this functional. The pre-image of any field element under a linear functional is an affine subspace of codimension one, which means there are $q^{2m-1}$ possibilities. The number of possibilities for the first pair are thus $(q^{2m}-1)(q^{2m-1})$.

Once the first pair is chosen, all the remaining pairs must be chosen from the $(2m-2)$-dimensional subspace orthogonal to its span, and within that space, everything proceeds exactly the same way as if w were solving the original problem for $2(m-1)$ instead of $2m$. In other words, we have:

$|Sp(2m,q)|=(q^{2m}-1)(q^{2m-1})|Sp(2(m-1),q)|$

Carrying through the induction, and noting that $|Sp(0,q)|=1$, we have:

$|Sp(2m,q)|=\prod _{i=1}^{m}[(q^{2i}-1)(q^{2i-1})]$

Collecting the powers of $q$, we note that the exponents add up to $m^{2}$, and we get:

$|Sp(2m,q)|=q^{m^{2}}\prod _{i=1}^{m}(q^{2i}-1)$

Particular cases

$m$	$2m$	Order formula for $Sp(2m,q)$
1	2	$q(q^{2}-1)=q^{3}-q$
2	4	$q^{4}(q^{2}-1)(q^{4}-1)$
3	6	$q^{9}(q^{2}-1)(q^{4}-1)(q^{6}-1)$