> [!NOTE] Theorem ($\text{Eucl}_{2}(\mathbb{R})$ is indeed a group) > The [[Second Euclidean Group]] $(\text{Eucl}(\mathbb{R}^{2}), \circ)$ is a [[Groups|group]]. *Proof*. $(G0)$ Follows from [[Isometry of The Plane#^1c0497|composition of plane isometries is a plane isometry]]. $(G1)$ Follows from [[Function Composition#^f4ab9a|associativity of function composition]]. $(G2)$ Cleary the [[Identity Function|identity map]] on $\mathbb{R}^{2}$ is an isometry on $\mathbb{R}^{2}$ that fixes every point, including the origin. $(G3)$ Take $f\in \text{Eucl}(\mathbb{R}^{2}).$ The inverses of the two [[Isometry of The Plane#^e4939d|types]] are $f^{-1}(z)=e^{-i\theta }(z-w)$ and $f^{-1}(z)=\overline{e^{-i\theta }(z-w)}=e^{i\theta}\bar{z}-e^{i\theta}\bar{w}.$