> [!NOTE] Definition ($\text{Eucl}(\mathbb{R}^{2})$) > The second [[Euclidean Group|Euclidean group]], denoted $\text{Eucl}(\mathbb{R}^{2}),$ is the set of all [[Isometry of The Plane|isometries of the plane]] under function composition. # Properties By [[Second Euclidean Group is Group]], $\text{Eucl}(\mathbb{R}^{2})$ is indeed a group. **Subgroups**: Note that [[Second Orthogonal Group Over The Reals|orthogonal group of the plane]], $O_{2}(\mathbb{R}),$ and the [[Second Orthogonal Group Over The Reals|special orthogonal group of the plane]] are subgroups of $\text{Eucl}(\mathbb{R}^{2}).$