\section{Properties of Cyclic Groups}

\begin{theorem}[Criterion for $\mathbf{a^i=a^j}$]
	Let $G$ be a group, and let $a$ belong to $G$. If $a$ has infinite order, then $a^i = a^j$ if and only if $i = j$. If $a$ has finite order, say, $n$, then $\lr{a} = \{e, a, a^2, \dots, a^{n - 1}\}$ and $a^i = a^j$ if and only if $n$ divides $i - j$.
\end{theorem}

\begin{corollary}[$\mathbf{\abs{a}=\abs{\lr{a}}}$]
	For any group element $a$, $\abs{a} = \abs{\lr{a}}$.
\end{corollary}

\begin{corollary}[$\mathbf{a^k = e}$ Implies That $\mathbf{\abs{a}}$ Divides $\mathbf{k}$]
	Let $G$ be a group and let $a$ be an element of order $n$ in $G$. If $a^k = e$, then $n$ divides $k$.
\end{corollary}

\begin{theorem}[$\mathbf{\lr{a^k} = \lr{a^{\textbf{gcd}(n,k)}}}$ and $\mathbf{\abs{a^k} = n/\textbf{gcd}(n,k)}$]
	Let $a$ be an element of order $n$ in a gruop and let $k$ be a positive integer. Then $\lr{a^k} = \lr{a^{\gcd(n,k)}}$ and $\abs{a^k} = n / \gcd(n,k)$.
\end{theorem}

\begin{corollary}[Orders of Elements in Finite Cyclic Groups]
	In a finite cyclic group, the order of an element divides the order of the group.
\end{corollary}

\begin{corollary}[Criterion for $\mathbf{\lr{a^i} = \lr{a^j}}$ and $\mathbf{\abs{a^i} = \abs{a^j}}$]
	Let $\abs{a} = n$. Then $\lr{a^i} = \lr{a^j}$ if and only if $\gcd(n, i) = \gcd(n,j)$, and $\abs{a^i} = \abs{a^j}$ if and only if $\gcd(n,i) = \gcd(n,j)$.
\end{corollary}

\begin{corollary}[Generators of Finite Cyclic Groups]
	Let $\abs{a} = n$. Then $\lr{a} = \lr{a^j}$ if and only if $\gcd(n,j) = 1$, and $\abs{a} = \abs{\lr{a^j}}$ if and only if $\gcd(n,j) = 1$.
\end{corollary}

\begin{corollary}[Generators of $\mathbf{\Z_n}$]
	An integer $k$ in $\Z_n$ is a generator of $Z_n$ if and only if $\gcd(n,k) = 1$.
\end{corollary}
