Cyclic Subgroup

Let $G$ be a group and let $g\in G$. The cyclic subgroup generated by $g$ is $\langle g\rangle \;=\;\{g^n : n\in\mathbb{Z}\},$ where powers are defined by $g^0=e$, $g^{n+1}=g^n g$ for $n\ge 0$, and $g^{-n}=(g^{-1})^n$ for $n\ge 1$. This is a subgroup of $G$. More generally, $\langle g\rangle$ is a special case of the subgroup generated by a subset .

A group is called cyclic if it equals $\langle g\rangle$ for some element $g$.

Examples:

• In $(\mathbb{Z},+)$, $\langle 3\rangle = 3\mathbb{Z}$.
• In $\mathbb{C}^{\times}$, $\langle i\rangle = \{1,i,-1,-i\}$.
• In $S_3$, $\langle (123)\rangle=\{e,(123),(132)\}$.
• $\langle e\rangle$ is the trivial subgroup.