Let be a group with identity , and let .

• The of , denoted , is the cardinality of .
• The of , denoted , is the smallest positive integer such that , if such an exists. If no such exists, we say has infinite order.

• None.

definition.order_for_groups.tex

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\begin{definition}
Let $G$ be a group with identity $e$, and let $g\in G$. 
\begin{itemize}
\item The \textdef{order} of $G$, denoted $|G|$, is the cardinality of $G$.
\item The \textdef{order} of $g$, denoted $|g|$, is the smallest positive integer $n$ such that $g^n = e$, if such an $n$ exists. If no such $n$ exists, $g$ is said to have infinite order.
\end{itemize}
\end{definition}

• Order (group theory)