\section{Terminology and Notation}

\begin{definition}[Order of a Group]
	The number of elements of a group (finite or infinite) is called its \textit{order}. We will use $\abs{G}$ to denote the order of $G$.
\end{definition}

\begin{definition}[Order of an Element]
	The \textit{order} of an element $g$ in a group $G$ is the smallest positive integer $n$ such that $g^n = e$. (In additive notation, this would be $ng = 0$.) If no such integer exists, we say that $g$ has \textit{infinite order}. The order of an element $g$ is denoted by $\abs{g}$.
\end{definition}

\begin{definition}[Subgroup]
	If a subset $H$ of a group $G$ is itself a group under the operation of $G$, we say that $H$ is a \textit{subgroup} of $G$.
\end{definition}