11 lines
540 B
TeX
11 lines
540 B
TeX
\section{Burnside's Theorem}
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\begin{definition}[Elements Fixed by $\mathbf{\phi}$]
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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 $\phi$").
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\end{definition}
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\begin{theorem}[Burnside's Theorem]
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If $G$ is a finite group of permutations on a set $S$, then the number of orbits of elements of $S$ under $G$ is
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\[ \frac{1}{\abs{G}}\sum_{\phi \in G}\abs{\fix(\phi)} \]
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\end{theorem}
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