Is the orthogonal group $O_n(\mathbb{R})$ isomorphic to $SO_n(\mathbb{R}) \times \{ \pm 1 \}$?

Question

Is the orthogonal group $O_n(\mathbb{R})$ isomorphic to $SO_n(\mathbb{R}) \times \{ \pm 1 \}$?

Here, $SO_n(\mathbb{R})$ is the set of orthogonal matrices with determinant $1$.

Answer 1

As in this question, there is a short exact sequence

$$1 \to SO(n) \to O(n) \xrightarrow{\det} \mathbb{Z}_2 \to 1$$

which splits, allowing us to write $O(n)$ as a semidirect product $SO(n) \rtimes \mathbb{Z}_2$. This semidirect product is nontrivial when $n$ is even (because there is no splitting $\mathbb{Z}_2 \to O(n)$ whose image commutes with $SO(n)$) but trivial when $n$ is odd, with the splitting given by sending $-1 \in \mathbb{Z}_2$ to $-1 \in O(n)$.