# Stabilizer of a nonsingular vector in a quadratic space (char (k)=2)

suppose that $k$ is a finite field of characteristic 2 and $(V,q)$ a quadratic space, i.e., $V$ is a $k$-vector space and $q:V\to k$ quadratic form. Suppose that $\dim(V)\geq 4$ and that $q$ is non-defective, i.e., the $k$-bilinear $B(x,y)=q(x+y)-q(x)-q(y)$ is non-degenerated:

The question is:

Given a non-singular vector $0\neq v\in V$, (that is $q(v)\neq 0$), What is the isomorphism type of the stabilizer of $v$ in $O'(V,q)$?

Note: I know the answer in the case that $k=\mathbb F_2$. In this case the stabilizer is $O(2n-1,2)\cong Sp(2n-2,2)$, where $2n=\dim(V)$ (but I do not have a reference). I suppose that the same is true in general, that is the stabilizer of a non-singular vector is isomorphic to $Sp(2n-2,k)$.

Thanks.

• I like Classical Groups and geometric Algebra by Larry C. Grove, chapters devoted to characteristic 2 Mar 13, 2014 at 1:39
• Not sure he does your question, though, after reading those bits. Mar 13, 2014 at 1:48
• For any $k$ of char. 2, $B_q$ on the hyperplane $H=v^{\perp}$ has 1-dimensional defect space $L$, with symplectic form $\overline{B}$ on $H/L$. Also, $G:={\rm{Stab}}_v(O(q))$ has 2 connected components, each geometrically connected over $k$, and $G^0 \rightarrow{\rm{Sp}}(\overline{B})\simeq{\rm{Sp}}_{2n-2}$ is a $k$-group isomorphism. Indeed, WLOG $k = \overline{k}$, so WLOG $q(v)=1$, and then use the self-contained proof of Prop. C.3.1 of math.stanford.edu/~conrad/papers/luminysga3.pdf. For finite $k$ this proof gives $G(k)={\rm{Sp}}_{2n-2}(k)\times\mathbf{Z}/(2)$ by Lang's theorem. Mar 13, 2014 at 5:30

The notation used for the various orthogonal groups varies a lot from book to book, so I am not completely sure what you mean by $O(n,q)$, but presumably $O'(n,q)$ is intended to denote the simple subgroup, which is often denoted $\Omega(n,q)$ or $\Omega_n(q)$.
The stabilizer of a non-singular $1$-space in the simple group $\Omega^{\pm}_{2n}(q)$ is isomorphic to ${\rm Sp}_{2n-2}(q)$. There is a unique conjugacy class of subgroups of this type, so it is normalized by the diagonal outer automorphism of $\Omega^{\pm}_{2n}(q)$, and the stabilizer in ${\rm SO}^{\pm}_{2n}(q) = {\rm GO}^{\pm}_{2n}(q)$ is isomorphic to ${\rm Sp}_{2n-2}(q) \times C_2$.
You asked about the stabilizer of a vector rather than a $1$-space, but that will be the same, because there are no scalars of order $2$ when $q$ is even.