Let $G$ be a 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, $g$ is said to have infinite order.

• None.

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}

• Order (group theory)

definition needsreview rjosh