\section{Applications of Factor Groups}

\begin{theorem}[$\mathbf{G/Z}$ Theorem]
	Let $G$ be a group and let $Z(G)$ be the center of $G$. If $G/Z(G)$ is cyclic, then $G$ is Abelian.
\end{theorem}

\begin{theorem}[$\mathbf{G/Z(G) \approx \text{Inn}(G)}$]
	For any group $G$, $G/Z(G)$ is isomorphic to Inn$(G)$.
\end{theorem}

\begin{theorem}[Cauchy's Theorem for Abelian Groups]
	Let $G$ be a finite Abelian group and let $p$ be a prime that divides the order of $G$. Then $G$ has an element of order $p$.
\end{theorem}
