> [!NOTE] Definition ($O_{2}(\mathbb{R})$) > The second [[Orthogonal Group|orthogonal group]] on $\mathbb{R}$ (*or the orthogonal group on* $\mathbb{R}^{2}$), denoted $O_{2}(\mathbb{R})$ or $O(2,\mathbb{R}),$ is the set of [[Isometry of The Plane|isometries of the plane]] that fix the origin under [[Function Composition|function composition]]. > >Equivalently, we may define the set of $O_{2}(\mathbb{R})$ as the following subset of the [[General Linear Group|general linear group]] $\{ M\in \text{GL}_{2}(\mathbb{R}) \mid M^{T} = M^{-1} \}$ ........ # Properties By [[Second Orthogonal Group Over The Reals is Group]], $O_{2}(\mathbb{R})$ is indeed a group under function composition. By [[Second Orthogonal Group Over The Reals is Isomorphic to First Orthogonal Group Over Complex Numbers]], $O_{2}(\mathbb{R}) \cong O_{1}(\mathbb{C}).$ **Subgroups**: The [[Second Special Orthogonal Group Over The Reals|second special orthogonal group over the reals]] which is the set of all rotations on $\mathbb{R}^{2},$ denoted $SO_{2}(\mathbb{R}),$ is a subgroup of $O_{2}(\mathbb{R}).$