A [[Cyclic Group]] (or monogeneuous group) of order $n$, denoted as $C_n$, is defined as a [[Group]] $(G,\circ)$ which is generated by a single element $g$ such that every element in the [[Set|set]] $G$ can be expressed as a [[Functional Powers|power]] of $g$, where the $n$-th power refers to $g^{n}=\underbrace{g \circ g \circ \cdots}_{n\text{{ times}}}$). >[!example] >The symmetric rotations of a $n$-sided [[Regular Polygon]] form a cyclic group, as its almost identical to a [[Dihedral Group]] of order $n$ but without [[Reflection Transformations|reflections]]. Every [[Cyclic Group]] that is also an [[Infinite Group]] is [[Isomorphism|isomorphic]] to [[Additive Group]] of [[Integer|integers]] $\Z$, while a [[Cyclic Group]] that is [[Finite Group|finite]] of order $n$ is [[Isomorphism|isomorphic]] to the [[Additive Group]] of $\Z$ [[Modular Arithmetic|mod]] $n$. Note that for an [[Infinite Group|infinite group]] $(G, *)$ to be cyclic, $G$ must be [[Infinite Set|Countably Infinite Set]], so the group of rotational symmetries of a [[Circle]] would not form a [[Cyclic Group]].