I am reading a claim that Og , the orthogonal group associated with a finite-dimensional vector space V over $\mathbb Z_2$ , and a quadratic form q defined therein , i.e., the group of linear transformations T:V-->V with q(v)=q(T(v)), is finitely-generated (and its generators are known) , because of the Cartan-Dieudonne' theorem. Now, the Cartan-Dieudonne theorem states that if V is a vector space of dimension n, then n is generated by at most n reflections. Now, my confusion here is that , since we are working over $\mathbb Z_2$ , orthogonal transformations are also symplectic, and the symplectic group is known to be finitely-generated by transvections (the generalization to vector spaces of shear maps in $\mathbb R^n$). Anyway, I am trying to figure out a generating set for Og as defined. Any Ideas? Thanks.