Orthogonal Group is Subgroup of General Linear Group

Theorem

Let $k$ be a field.

Let $\map {\operatorname O} {n, k}$ be the $n$th orthogonal group on $k$.

Let $\GL {n, k}$ be the $n$th general linear group on $k$.

Then $\map {\operatorname O} {n, k}$ is a subgroup of $\GL {n, k}$.

Proof

From Unit Matrix is Orthogonal, the unit matrix $\mathbf I_n$ is orthogonal.

Let $\mathbf A, \mathbf B \in \map {\operatorname O} {n, k}$.

Then, by definition, $\mathbf A$ and $\mathbf B$ are orthogonal.

Then by Inverse of Orthogonal Matrix is Orthogonal:

$\mathbf B^{-1}$ is a orthogonal matrix.

By Product of Orthogonal Matrices is Orthogonal Matrix:

$\mathbf A \mathbf B^{-1}$ is a orthogonal matrix.

Thus by definition of orthogonal group:

$\mathbf A \mathbf B^{-1} \in \map {\operatorname O} {n, k}$

Hence the result by One-Step Subgroup Test.

$\blacksquare$