Created the Abstract Algebra theorems and definitions cheat sheet

part-2/chapters/chapter-10/chapter-10.tex

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+\chapter{Group Homomorphisms}
+\subimport{./}{definition-and-examples.tex}
+\subimport{./}{properties-of-homomorphisms.tex}
+\subimport{./}{the-first-isomorphism-theorem.tex}

part-2/chapters/chapter-10/definition-and-examples.tex

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+\section{Definition and Examples}
+
+\begin{definition}[Group Homomorphism]
+	A \textit{homomorphism} $\phi$ from a group $G$ to a group $\overline{G}$ is a mapping from $G$ into $\overline{G}$ that preserves the group operation; that is, $\phi(ab) = \phi(a)\phi(b)$ for all $a, b$ in $G$.
+\end{definition}
+
+\begin{definition}[Kernel of a Homomorphism]
+	The \textit{kernel} of a homomorphism $\phi$ from a group $G$ to a group with identity $e$ is the set $\{x \in G\ \vert\ \phi(x)=e\}$. The kernel of $\phi$ is denoted by $\ker\phi$.
+\end{definition}

part-2/chapters/chapter-10/properties-of-homomorphisms.tex

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+\section{Properties of Homomorphisms}
+
+\begin{theorem}[Properties of Elements Under Homomorphisms]
+	Let $\phi$ be a homomorphism from a group $G$ to a group $\overline{G}$ and let $g$ be an element of $G$. Then
+	\begin{enumerate}
+		\item $\phi$ carries the identity of $G$ to $\overline{G}$.
+		\item $\phi(g^n)=(\phi(g))^n$ for all $n$ in $\Z$.
+		\item If $\abs{g}$ is finite, then $\abs{\phi(g)}$ divides $\abs{g}$.
+		\item $\ker\phi$ is a subgroup of $G$.
+		\item $\phi(a) = \phi(b)$ if and only if $a\ker\phi = b\ker\phi$.
+		\item If $\phi(g) = g'$, then $\phi^{-1}(g') = \{x \in G\ \vert\ \phi(x) = g'\} = g\ker\phi$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[Properties of Subgroups Under Homomorphisms]
+	Let $\phi$ be a homomorphism from a group $G$ to a group $\overline{G}$ and let $H$ be a subgroup of $G$. Then
+	\begin{enumerate}
+		\item $\phi(H) = \{\phi(h)\ \vert\ h \in H\}$ is a subgroup of $\overline{G}$.
+		\item If $H$ is cyclic, then $\phi(H)$ is cyclic.
+		\item If $H$ is Abelian, then $\phi(H)$ is Abelian.
+		\item If $H$ is normal in $G$, then $\phi(H)$ is normal in $\phi(G)$.
+		\item If $\abs{\ker\phi} = n$, then $\phi$ is an $n$-to-1 mapping from $G$ onto $\phi(G)$.
+		\item If $\abs{H} = n$, then $\abs{\phi(H)}$ divides $n$.
+		\item If $\overline{K}$ is a subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K})=\{k \in G\ \vert\ \phi(k) \in \overline{K}\}$ is a subgroup of $G$.
+		\item If $\overline{K}$ is a normal subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K})=\{ k \in G\ \vert\ \phi(k) \in \overline{K}\}$ is a normal subgroup of $G$.
+		\item If $\phi$ is onto and $\ker\phi = \{e\}$, then $\phi$ is an isomorphism from $G$ to $\overline{G}$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}[Kernels Are Normal]
+	Let $\phi$ be a group homomorphism from $G$ to $\overline{G}$. Then $\ker\phi$ is a normal subgroup of $G$.
+\end{corollary}

part-2/chapters/chapter-10/the-first-isomorphism-theorem.tex

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+\section{The First Isomorphism Theorem}
+
+\begin{theorem}[First Isomorphism Theorem (Jordan, 1870)]
+	Let $\phi$ be a group homomorphism from $G$ to $\overline{G}$. Then the mapping from $G/\ker\phi$ to $\phi(G)$, given by $g\ker\phi \to \phi(g)$, is an isomorphism. In symbols, $G/\ker\phi \approx \phi(G)$.
+\end{theorem}
+
+\begin{corollary}
+	If $\phi$ is a homomorphism from a finite group $G$ to $\overline{G}$, then $\abs{\phi(G)}$ divides $\abs{G}$ and $\abs{\overline{G}}$.
+\end{corollary}
+
+\begin{theorem}[Normal Subgroups Are Kernels]
+	Every normal subgroup of a group $G$ is the kernel of a homomorphism of $G$. In particular, a normal subgroup $N$ is the kernel of the mapping $g \to gN$ from $G$ to $G/N$.
+\end{theorem}