Created the Abstract Algebra theorems and definitions cheat sheet

part-5/chapters/chapter-26/chapter-26.tex

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+\chapter{Generators and Relations}
+\subimport{./}{motivation.tex}
+\subimport{./}{definitions-and-notation.tex}
+\subimport{./}{free-group.tex}
+\subimport{./}{generators-and-relations.tex}
+\subimport{./}{classification-of-groups-of-order-up-to-15.tex}
+\subimport{./}{characterization-of-dihedral-groups.tex}

part-5/chapters/chapter-26/characterization-of-dihedral-groups.tex

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+\section{Characterization of Dihedral Groups}
+
+\begin{theorem}[Characterization of Dihedral Groups]
+	Any group generated by a pair of elements of order 2 is dihedral.
+\end{theorem}

part-5/chapters/chapter-26/classification-of-groups-of-order-up-to-15.tex

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+\section{Classification of Groups of Order Up to 15}
+
+\begin{theorem}[Classification of Groups of Order 8 (Cayley, 1859)]
+	Up to isomorphism, there are only five groups of order 8: $\Z_8$, $\Z_4 \oplus \Z_2$, $\Z_2 \oplus \Z_2 \oplus \Z_2$, $D_4$, and the quaternions.
+\end{theorem}

part-5/chapters/chapter-26/definitions-and-notation.tex

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+\section{Definitions and Notation}
+
+\begin{remark}
+	For any set $S=\{a,b,c,\dots\}$ of distinct symbols, we create a new set $S^{-1} = \{a^{-1},b^{-1},c^{-1},\dots\}$ by replacing each $x$ in $S$ by $x^{-1}$. Define the set $W(S)$ to be the collection of all formal finite strings of the form $x_1x_2\dots x_k$, where each $x_i \in S \cup S^{-1}$. The elements of $W(S)$ are called \textit{words from $S$}. We also permit the string with no elements to be in $W(S)$. this word is called the \textit{empty word} and is denoted by $e$.
+
+	We may define a binary operation on the set $W(S)$ by juxtaposition; that is, if $x_1x_2\dots x_k$ and $y_1y_2\dots y_t$ belong to $W(S)$, then so does $x_1x_2\dots x_ky_1y_2\dots y_t$. Observe that this operation is associative and the empty word is the identity. Also, notice that a word such as $aa^{-1}$ is not the identity, because we are treating the elements of $W(S)$ as formal symbols with no implied meaning.
+\end{remark}
+
+\begin{definition}[Equivalence Classes of Words]
+	For any pair of elements $u$ and $v$ of $W(S)$, we say that $u$ is related to $v$ if $v$ can be obtained from $u$ by a finite sequence of insertions or deletions of words of the form $xx^{-1}$ or $x^{-1}x$, where $x \in S$.
+\end{definition}

part-5/chapters/chapter-26/free-group.tex

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+\section{Free Group}
+
+\begin{theorem}[Equivalence Classes Form a Group]
+	Let $S$ be a set of distinct symbols. For any word $u$ in $W(S)$, let $\overline{u}$ denote the set of all words in $W(S)$ equivalent to $u$ (that is, $\overline{u}$ is the equivalence class containing $u$). Then the set of all equivalence classes of elements of $W(S)$ is a group under the operation $\overline{u}\cdot\overline{v} = \overline{uv}$. This group is called a \textit{free group on $S$}.
+\end{theorem}
+
+\begin{theorem}[Universal Mapping Property]
+	Every group is a homomorphic image of a free group.
+\end{theorem}
+
+\begin{corollary}[Universal Factor Group Property]
+	Every group is isomorphic to a factor group of a free group.
+\end{corollary}

part-5/chapters/chapter-26/generators-and-relations.tex

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+\section{Generators and Relations}
+
+\begin{definition}[Generators and Relations]
+	Let $G$ be a group generated by some subset $A=\{a_1,a_2,\dots,a_n\}$ and let $F$ be the free group on $A$. Let $W = \{w_1,w_2,\dots,w_t\}$ be a subset of $F$ and let $N$ be the smallest normal subgruop of $F$ containing $W$. We say that $G$ is \textit{given by the generators $a_1,a_2,\dots,a_n$ and the relations $w_1=w_2=\dots=w_t=e$} if there is an isomorphism from $F/N$ onto $G$ that carries $a_iN$ to $a_i$.
+
+	\noindent The notation for this situation is
+	\[ G = \lr{a_1,a_2,\dots,a_n\ \vert\ w_1=w_2=\dots=w_t=e} \]
+\end{definition}
+
+\begin{theorem}[Dyck's Theorem (1882)]
+	Let
+	\[ G = \lr{a_1,a_2,\dots,a_n\ \vert\ w_1=w_2=\dots=w_t=e} \]
+	and let
+	\[ \overline{G}=\lr{a_1,a_2,\dots,a_n\ \vert\ w_1=w_2=\dots=w_t=w_{t+1}=\dots=w_{t+k}=e} \]
+	Then $\overline{G}$ is a homomorphic image of $G$.
+\end{theorem}
+
+\begin{corollary}[Largest Group Satisfying Defining Relations]
+	If $K$ is a group satisfying the defining relations of a finite group $G$ and $\abs{K} \geq \abs{G}$, then $K$ is isomorphic to $G$.
+\end{corollary}

part-5/chapters/chapter-26/motivation.tex

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+\section{Motivation}
+
+\begin{remark}
+	In this chapter, we present a convenient way to define a group with certain prescribed properties. Simply put, we begin with a set of elements that we want to generate the group, and a set of equations (called \textit{relations}) that specify the conditions that these generators are to satisfy. Among all such possible groups, we will select one that is as large as possible. This will uniquely determine the group up to isomorphism.
+\end{remark}