Definition:Orthogonal Group

Definition
Let $k$ be a field.

The ($n$th) orthogonal group (on $k$), $\operatorname O \left({n, k}\right)$, is the following subset of the general linear group $\operatorname{GL} \left({n, k}\right)$:


 * $\operatorname O \left({n, k}\right) := \left\{{M \in \operatorname{GL} \left({n, k}\right): M^t = M^{-1}}\right\}$

where $M^t$ denotes the transpose of $M$.

Further, $\operatorname O \left({n, k}\right)$ is considered to be endowed with conventional matrix multiplication.