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Let $q$ be a power of $2$ and let $(V,Q)$ be a quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type and the minus type. The plus type corresponds to the case where a maximal isotropic subspace of $V$ has dimension $m$ and the minus type corresponds to the case where a maximal isotropic subspace has dimension $m-1$. Let us denote by $O_{2m}^{+}(q)$ (resp. $O_{2m}^{-}(q)$) the group of isometries of $(V,Q)$ when $Q$ is of plus type (resp. minus type).

Q1: What is a sharp upper bound (or the exact value) for the for the maximal order of an element inside $O_{2m}^{\pm}(q)$ ?

Q2 What is the shape of the matrices (in a convenient chosen basis ) which represent the elements in Q1 ?

added If we define the bilinear form $$ f(x,y):=Q(x+y)+Q(x)+Q(y) $$ then because we are in characteristic $2$ we get that $f$ is alternating and therefore we can think of the orthogonal group as being a subgroup of a symplectic group. Therefore we can try to bound the order of elements in $Sp(2m,q)$.

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One paper I found, is a recent preprint "Spectra of finite symplectic and orthogonal groups" by Buturlakin. Spectra means the set of orders here. It is unclear which parts are for odd $q$ and which even, but Corollary 4 (page 29) seems to give the full spectrum in the case you desire. – Junkie Aug 31 '11 at 7:39

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