----- > [!proposition] Proposition. ([[finite subgroups of O2 are cyclic or dihedral]]) > The finite subgroups of $O_{2}$ are, up to [[group isomorphism|isomorphism]] the [[cyclic group|cyclic]] and [[dihedral group|dihedral groups]] of $n \in \mathbb{N}$. > [!proof]- Proof. ([[finite subgroups of O2 are cyclic or dihedral]]) > > $\begin{align} > \text{SO}_{2} \ & \trianglelefteq & \text{O}_{2} \\ > \cup & & \cup \\ > \underbrace{G \cap \text{SO}_{2} }_{H} & \trianglelefteq & G > \end{align}$ > > There is a natural map $G \xrightarrow{\det} \text{O}_{2} / \text{SO}_{2}$ with $\ker \det = H$ and so by the [[first isomorphism theorem]] we get an [[injection]] $G / H \hookrightarrow \text{O}_{2} / \text{SO}_{2} \cong \{ \pm 1 \}$. The finite [[subgroup]]s of $\text{SO}_{2}$ are [[group isomorphism|isomorphic]] to $C_{n}$ for some $n$ (since they're generated by rotations by $2\pi / n$). So $H \cong C_{n}$ has [[index of a subgroup|index]] $1$ or $2$ in $G$. If $1$ then immediately $H \cong G$. Else $2$ and there exists some $r' \notin H$. $r'$ must be a [[reflection]] due to the [[internal semi-direct product|semi-direct product]] structure of the [[group]] $\text{O}_{2}= \text{SO}_{2} \rtimes \langle r \rangle$ enforcing $G=H\langle r' \rangle$. Now, $r'\rho_{\theta}r'=\rho_{-\theta}=\rho_{\theta}^{-1}$ and thus the relations of the [[dihedral group]] $D_{n}$ are satisfied. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```