---- > [!definition] Definition. ([[cyclic group]]) > A [[group]] $G$ is said to be **cyclic** if it is [[generating set of a group|generated by]] a single element: $G=\langle a \rangle$ for some $a \in G$. > [!proposition] Proposition 1. > Every **cyclic group** is [[countably infinite|countable]] and [[group isomorphism|isomorphic]] to either $\mathbb{Z}$ or —if finite— to $\mathbb{Z} / n\mathbb{Z}$ for some $n \in \mathbb{N}$. As a **corollary**, any two [[cyclic group]]s of [[order of a group|order]] $n$ are [[group isomorphism|isomorphic]] ($G_{1}\cong \mathbb{Z}_{n}\cong G_{2}$). > \ > **Remark.** This means that for all $n \in \mathbb{N}$, there is a [[cyclic group]] of order $n$. > \ > This fact is so ubiquitous that we often will just say 'the cyclic groups are $\mathbb{Z} \text{ and }\mathbb{Z} / n\mathbb{Z}. > [!proposition] Proposition 2. > If $|G|=p$ for $p$ a [[prime number]], then $G$ is **cyclic**. > > [!proof] Proof of Proposition 1. > Suppose $G=\langle a \rangle$ is **cyclic** with $|G|=n$. Then one [[group isomorphism|isomorphism]] takes $a \mapsto 1,a^{2}\mapsto 2, \dots,a^{n-1} \mapsto n-1, a^{n}=e=0$, for then $a^{i}a^{j}=a^{i+j \text{ mod } n} \mapsto i+j \text{ mod } n.$ So $|G|\cong \mathbb{Z}_{n}.$ If $|G|$ is unbounded, then the [[group isomorphism|isomorphism]] is just $e_G \mapsto 0, a \mapsto 1, \dots, a_{m}\mapsto m \dots$. > [!proof] Proof of Proposition 2. > Let $a \in G$ and consider the [[subgroup]] $\langle a \rangle$. Since [[Lagrange's Theorem]] implies that the only nontrivial [[subgroup]] of $G$ is $G$ itself, we see $\langle a \rangle =G$. ---- #### ---- #### 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 > ```