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====== Order (for Groups) ======
====Definition==== 
Let $G$ be a [[definition:group]] with identity $e$, and let $g\in G$. 
  * The $\textdef{order}$ of $G$, denoted $|G|$, is the cardinality of $G$.
  * 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, we say $g$ has infinite order.

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==== Remarks ==== 
  * None.


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==== $\LaTeX$ version ====
<file tex definition.order_for_groups.tex>
%%%%%
% DEPENDENCIES 
% RequiredMacros: \newcommand{\textdef}[1]{\textit{#1}} 
%%%%%
\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}
</file>

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==== External links ====
  * [[wp>Order (group theory)]]


{{tag>definition needsreview rjosh rben}}

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