Created the Abstract Algebra theorems and definitions cheat sheet

part-5/chapters/chapter-30/chapter-30.tex

@@ -0,0 +1,3 @@
+\chapter{Cayley Digraphs of Groups}
+\subimport{./}{the-cayley-digraph-of-a-group.tex}
+\subimport{./}{hamiltonian-circuits-and-paths.tex}

part-5/chapters/chapter-30/hamiltonian-circuits-and-paths.tex

@@ -0,0 +1,20 @@
+\section{Hamiltonian Circuits and Paths}
+
+\begin{remark}
+	Obviously, this idea can be applied to any digraph; that is, one starts at some vertex and attempts to traverse the digraph by moving along arcs in such a way that each vertex is visited exactly once before returning to the starting vertex. (To go from $x$ to $y$, there must be an arc from $x$ to $y$.) Such a sequence of arcs is called a \textit{Hamiltonian circuit} in the digraph. A sequence of arcs that passes through each vertex exactly once without returning to the starting point is called a \textit{Hamiltonian path}. In the rest of this chapter, we concern ourselves with the existence of Hamiltonian circuits and paths in Cayley digraphs.
+\end{remark}
+
+\begin{theorem}[A Necessary Condition]
+	Cay$(\{(1,0),(0,1)\}:\Z_m \oplus \Z_n)$ does not have a Hamiltonian circuit when $m$ and $n$ are relatively prime and greater than 1.
+\end{theorem}
+
+\begin{theorem}[A Sufficient Condition]
+	Cay$(\{(1,0),(0,1)\}:\Z_m \oplus \Z_n)$ has a Hamiltonian circuit when $n$ divides $m$.
+\end{theorem}
+
+\begin{theorem}[Abelian Groups Have Hamiltonian Paths]
+	Let $G$ be a finite Abelian group, and let $S$ be any (nonempty*) generating set for $G$. Then Cay$(S:G)$ has a Hamiltonian path.\\
+
+
+	\noindent *If $S$ is the empty set, it is customary to define $\lr{S}$ as the identity group. We prefer to ignore this trivial case.
+\end{theorem}

part-5/chapters/chapter-30/the-cayley-digraph-of-a-group.tex

@@ -0,0 +1,9 @@
+\section{The Cayley Digraph of a Group}
+
+\begin{definition}[Cayley Digraph of a Group]
+	Let $G$ be a finite group and let $S$ be a set of generators for $G$. We define a digraph Cay$(S:G)$, called the \textit{Cayley digraph of $G$ with generating set $S$}, as follows.
+	\begin{enumerate}
+		\item Each element of $G$ is a vertex of Cay$(S:G)$.
+		\item For $x$ and $y$ in $G$, there is an arc from $x$ to $y$ if and only if $xs=y$ for some $s \in S$.
+	\end{enumerate}
+\end{definition}