\section{Burnside's Theorem}

\begin{definition}[Elements Fixed by $\mathbf{\phi}$]
	For any group $G$ of permutations on a set $S$ and any $\phi$ in $G$, we let $\fix(\phi) = \{i \in S\ \vert\ \phi(i)=i\}$. This set is called the \textit{elements fixed by $\phi$} (or more simply, "fix of $\phi$").
\end{definition}

\begin{theorem}[Burnside's Theorem]
	If $G$ is a finite group of permutations on a set $S$, then the number of orbits of elements of $S$ under $G$ is
	\[ \frac{1}{\abs{G}}\sum_{\phi \in G}\abs{\fix(\phi)} \]
\end{theorem}
