Let $G$ be a group with group operation written multiplicatively, let $e \in G$ denote its identity element, let $\mathbb{N}=\{1,2,3,\dots\}$, and let $g \in G$. For $n \in \mathbb{Z}$, define $g^0=e$, define $g^n$ for $n \in \mathbb{N}$ as the product of $n$ copies of $g$, and define $g^n=(g^{-1})^{-n}$ for $n<0$. Define
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\begin{align*}
\langle g \rangle := \{g^n : n \in \mathbb{Z}\}.
\end{align*}
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Then $\langle g \rangle \le G$. Moreover, if $H \le G$ and $g \in H$, then $\langle g \rangle \subset H$. Thus $\langle g \rangle$ is the smallest subgroup of $G$ containing $g$.