## Relationship Between Symplectic Group and Orthogonal Group

Hi, All:

Given a simplectic vector space (V,w), i.e., V is an n-dim. Vector Space ( n finite)

and w is a symplectic form, i.e., a bilinear, antisymmetric totally isotropic and

non-degenerate form, the simplectic groupSp(2n) of V is the (sub)group of GL(V) that

preserves this form. Similarly, given a bilinear, symmetric non-degenerate form q in V,

the orthogonal group O(n) is the subgroup of Gl(V) that preserves q.

Question: is there some relationship between these two groups under some conditions

, i.e containment, overlap, etc? I think the two groups agree when we work with Z/2-

coefficients (since 1=-1 implies that symmetry and antisymmetry coincide), but I am

clueless otherwise. I have gone thru Artin's geometric algebra, but I cannot get

a clear answer to the question.

Anyone know, or have a ref?