\begin{definition}[Definition:Orthogonal Group]
Let $k$ be a field.
The '''($n$th) orthogonal group (on $k$)''', denoted $\map {\mathrm O} {n, k}$, is the following subset of the general linear group $\GL {n, k}$:
:$\map {\mathrm O} {n, k} := \set {M \in \GL {n, k}: M^\intercal = M^{-1} }$
where $M^\intercal$ denotes the transpose of $M$.
Further, $\map {\mathrm O} {n, k}$ is considered to be endowed with conventional matrix multiplication.
That is, the '''($n$th) orthogonal group (on $k$)''' is the set of all orthogonal order-$n$ square matrices over $k$ under (conventional) matrix multiplication.
\end{definition}