Let $G$ be a group with identity element $e_G$, and let $G$ act on a set $X$ by an action map
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\begin{align*}
G \times X &\to X
\end{align*}
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\begin{align*}
(h,y) &\mapsto h \cdot y.
\end{align*}
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Let $g \in G$ have finite order $n \in \mathbb{N}$, meaning that $\operatorname{ord}(g)=n$ and $n$ is the least positive integer such that $g^n=e_G$. Let $x \in X$. Let
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\begin{align*}
\langle g \rangle = \{g^k : k \in \mathbb{Z}\}
\end{align*}
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denote the cyclic subgroup of $G$ generated by $g$. Then the orbit of $x$ under $\langle g \rangle$,
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\begin{align*}
\langle g \rangle \cdot x = \{h \cdot x : h \in \langle g \rangle\},
\end{align*}
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is finite, and
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\begin{align*}
|\langle g \rangle \cdot x| \mid n.
\end{align*}