Created the Abstract Algebra theorems and definitions cheat sheet

part-2/chapters/chapter-9/applications-of-factor-groups.tex

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+\section{Applications of Factor Groups}
+
+\begin{theorem}[$\mathbf{G/Z}$ Theorem]
+	Let $G$ be a group and let $Z(G)$ be the center of $G$. If $G/Z(G)$ is cyclic, then $G$ is Abelian.
+\end{theorem}
+
+\begin{theorem}[$\mathbf{G/Z(G) \approx \text{Inn}(G)}$]
+	For any group $G$, $G/Z(G)$ is isomorphic to Inn$(G)$.
+\end{theorem}
+
+\begin{theorem}[Cauchy's Theorem for Abelian Groups]
+	Let $G$ be a finite Abelian group and let $p$ be a prime that divides the order of $G$. Then $G$ has an element of order $p$.
+\end{theorem}

part-2/chapters/chapter-9/chapter-9.tex

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+\chapter{Normal Subgroups and Factor Groups}
+\subimport{./}{normal-subgroups.tex}
+\subimport{./}{factor-groups.tex}
+\subimport{./}{applications-of-factor-groups.tex}
+\subimport{./}{internal-direct-products.tex}

part-2/chapters/chapter-9/factor-groups.tex

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+\section{Factor Groups}
+
+\begin{theorem}[Factor Groups (O. Hölder, 1889)]
+	Let $G$ be a group and let $H$ be a normal subgroup of $G$. The set $G/H = \{ aH\ \vert\ a \in G\}$ is a group under the operation $(aH)(bH) = abH$.
+\end{theorem}

part-2/chapters/chapter-9/internal-direct-products.tex

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+\section{Internal Direct Products}
+
+\begin{definition}[Internal Direct Product of $\mathbf{H}$ and $\mathbf{K}$]
+	We say that $G$ is the \textit{internal direct product} of $H$ and $K$ and write $G = H \times K$ if $H$ and $K$ are normal subgroups of $G$ and
+	\[ G = HK\ \ \ \ \text{and}\ \ \ \ H \cap K = \{e\} \]
+\end{definition}
+
+\begin{definition}[Internal Direct Product $\mathbf{H_1 \times H_2 \times \dots \times H_n}$]
+	Let $H_1, H_2,\dots,H_n$ be a finite collection of normal subgroups of $G$. We say that $G$ is the \textit{internal direct product} of $H_1,H_2,\dots,H_n$ and write $G=H_1\times H_2 \times \dots \times H_n$, if
+	\begin{enumerate}
+		\item $G = H_1H_2\dots H_n = \{h_1h_2\dots h_n\ \vert\ h_i \in H_i\}$,
+		\item $(H_1H_2\dots H_n) \cap H_{i + 1} = {e}$ for $i=1,2,\dots, n-1$.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}[$\mathbf{H_1 \times H_2 \times \dots \times H_n \approx H_1 \oplus H_2 \oplus \dots \oplus H_n}$]
+	If a group $G$ is the internal direct product of a finite number of subgroups $H_1,H_2, \dots, H_n$, then $G$ is isomorphic to the external direct product of $H_1,H_2 \dots, H_n$.
+\end{theorem}
+
+\begin{theorem}[Classification of Groups of Order $\mathbf{p^2}$]
+	Every group of order $p^2$, where $p$ is a prime, is isomorphic to $\Z_{p^2}$ or $\Z_p \oplus \Z_p$.
+\end{theorem}
+
+\begin{corollary}
+	If $G$ is a group of order $p^2$, where $p$ is a prime, then $G$ is Abelian.
+\end{corollary}

part-2/chapters/chapter-9/normal-subgroups.tex

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+\section{Normal Subgroups}
+
+\begin{definition}[Normal Subgroup]
+	A subgroup $H$ of a group $G$ is called a \textit{normal} subgroup of $G$ if $aH = Ha$ for all $a$ in $G$. We denote this by $H \triangleleft G$.
+\end{definition}
+
+\begin{theorem}[Normal Subgroup Test]
+	A subgroup $H$ of $G$ is normal in $G$ if and only if $xHx^{-1} \subseteq H$ for all $x$ in $G$.
+\end{theorem}