TITLE: Orthogonal Groups over finite fields
QUESTION [13 upvotes]: Hello
Let $\mathbb{F}_p$ be the finite field with $p$ elements. One can show that over finite fields, there are just two non-degenerate quadratic forms. 
So here I want to pick 
any non-degenerate symmetric matrix $B$, and then look at the special orthogonal group defined by
$$
SO_{n}(\mathbb{F}_p,B):=\{ A\in SL_n(\mathbb{F}_p): ABA^T=B \}
$$
Is it true that the commutator subgroup of $SO_{n}$ is the whole group? In the other words I would like to know if $SO_{n}$ over finite fields is a perfect group.

REPLY [4 votes]: I think it's worth adding that there is a very detailed analysis of the orthogonal groups over arbitrary fields (not just finite fields, and including characteristic 2) in Dieudonné's "La Géométrie des Groupes Classiques". (The first edition of this book already goes back to 1954, but I'm referring to the 3rd edition from 1971.)
In particular, Chapter II, Section 8, seems to answer the question you're interested in when the characteristic is not 2, and Section 10 (in particular item (8) on p.67) answers your question when the characteristic is 2.