What is orthogonal group $O(1)$?

Question

I know that $O(2)$ is the group of 2x2 orthogonal matrices, but how can we extend the meaning of group and orthogonal to $O(1)$?

Answer 1

If $a\in\mathbb{R}^1$, $a\cdot a^T=1\Longrightarrow a^2=1\Longrightarrow a=\pm 1$. So $O(1)=\{\pm 1\}$.