# Generating Set for Og over $\mathbb Z_2$

Hi, All:

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.

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Never mind, I got it; I was looking at the wrong Dieudonne source. – Larry Jul 24 '11 at 22:21
It would be most convenient for us if you posted a brief explanation as an answer (so the software recognizes the question as resolved), or deleted the question. – S. Carnahan Jul 25 '11 at 1:34
Sure; the Dieudonne source is : La geommetrie des groupes clasiques (3rd. ed), Ergebnisse der Math u.i Grundz.5, springer, 1971. The maps are the $Z_2$ transvections (in this context) defined on H_1(Sg,Z/2) by: T_z(x):=x+(z,x)_2z, where (z,x)_2 is the intersection form in H_1(Sg,z/2) . Hope it helps anyone else interested. – Larry Jul 25 '11 at 2:16