Created the Abstract Algebra theorems and definitions cheat sheet

part-5/chapters/chapter-29/burnsides-theorem.tex

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+\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}

part-5/chapters/chapter-29/chapter-29.tex

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+\chapter{Symmetry and Counting}
+\subimport{./}{motivation.tex}
+\subimport{./}{burnsides-theorem.tex}
+\subimport{./}{group-action.tex}

part-5/chapters/chapter-29/group-action.tex

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+\section{Group Action}
+
+\begin{remark}
+	Our informal approach to counting the number of objects that are considered nonequivalent can be made formal as follows. If $G$ is a group and $S$ is a set of objects, we say that $G$ \textit{acts on} $S$ if there is a homomorphism $\gamma$ from $G$ to sym$(S)$, the group of all permutations on $S$. (The hommomorphism is sometimes called the \textit{group action}.) For convenience, we denote the image of $g$ under $\gamma$ as $\gamma_g$. Then two objects $x$ and $y$ in $S$ are viewed as equivalent under the action of $G$ if and only if $\gamma_g(x) = y$ for some $g$ in $G$. Notice that when $\gamma$ is one-to-one, the elements of $G$ may be regarded as permutations on $S$. On the other hand, when $\gamma$ is not one-to-one, the elements of $G$ may still be regarded as permutations on $S$, but there are distinct elements $g$ and $h$ in $G$ such that $\gamma_g$ and $\gamma_h$ induce the same permutation on $S$ [that is, $\gamma_g(x) = \gamma_h(x)$ for all $x$ in $S$]. Thus, a group acting on a set is a natural generalization of the permutation group concept.
+\end{remark}

part-5/chapters/chapter-29/motivation.tex

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+\section{Motivation}
+
+\begin{remark}
+	In general, we say that two designs (arrangements of beads) $A$ and $B$ are \textit{equivalent under a group $G$} of permutations of the arrangements if there is an element $\phi$ in $G$ such that $\phi(A) = B$. That is, two designs are equivalent under $G$ if they are in the same orbit of $G$. It follows, then, that the number of nonequivalent designs under $G$ is simply the number of orbits of designs under $G$. (The set being permuted is the set of all possible designs or arrangements.)
+\end{remark}