Cyclic Group

A cyclic group is defined by a group $G$ with operation $\ast$. We say that $G$ is a cyclic group if it can be generated by a generator element $g\in G:g^n=\underset{n\text{ times}}{\underbrace{g\ast g\ast \cdots \ast g}}$, where $g^0=e_G$ and $g^{-n}=(g^n{)}^{-1}=(g^{-1}{)}^n$. We write this as $⟨g⟩\le G$, such that $|⟨g⟩|=o(g)$.