Definition:Special Orthogonal Group

Definition
Let $k$ be a field.

The special ($n$th) orthogonal group (on $k$), denoted $\operatorname {SO} \left({n, k}\right)$, is:
 * the set of all proper orthogonal order-$n$ square matrices over $k$
 * under (conventional) matrix multiplication.

That is: $\operatorname{SO}_n(k) = \operatorname{O}_n(k) \cap \operatorname{SL}_n(k)$

Also see

 * Definition:Orthogonal Group
 * Definition:Special Unitary Group
 * Special Orthogonal Group is Group
 * Special Orthogonal Group is Subgroup of Orthogonal Group