Created the Abstract Algebra theorems and definitions cheat sheet

part-2/chapters/chapter-6/automorphisms.tex

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+\section{Automorphisms}
+
+\begin{definition}[Automorphism]
+	An isomorphism from a group $G$ onto itself is called an \textit{automorphisms} of $G$.
+\end{definition}
+
+\begin{definition}[Inner Automorphism Induced by $\mathbf{a}$]
+	Let $G$ be a group, and let $a \in G$. The function $\phi_a$ defined by $\phi_a(x) = axa^{-1}$ for all $x$ in $G$ is called the \textit{inner automorphism of $G$ induced by $a$}.
+\end{definition}
+
+\begin{theorem}[Aut($G$) and Inn($G$) Are Groups]
+	The set of automorphisms of a group and the set of inner automorphisms of a group are both groups under the operation of function composition.
+
+	When $G$ is a group, we use Aut($G$) to denote the set of all automorphisms of $G$ and Inn($G$) to denote the set of all inner automorphisms of $G$.
+\end{theorem}
+
+\begin{theorem}[Aut$\mathbf{(\Z_n) \approx U(n)}$]
+	For every positive integer $n$, Aut($\Z_n$) is isomorphic to $U(n)$.
+\end{theorem}

part-2/chapters/chapter-6/cayleys-theorem.tex

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+\section{Cayley's Theorem}
+
+\begin{theorem}[Cayley's Theorem (1854)]
+	Every group is isomorphic to a group of permutations.
+\end{theorem}

part-2/chapters/chapter-6/chapter-6.tex

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+\chapter{Isomorphisms}
+\subimport{./}{definition-and-examples.tex}
+\subimport{./}{cayleys-theorem.tex}
+\subimport{./}{properties-of-isomorphisms.tex}
+\subimport{./}{automorphisms.tex}

part-2/chapters/chapter-6/definition-and-examples.tex

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+\section{Definition and Examples}
+
+\begin{definition}[Group Isomorphism]
+	An \textit{isomorphism} $\phi$ from a group $G$ to a group $\overline{G}$ is a one-to-one mapping (or function) from $G$ onto $\overline{G}$ that preserves the group operation. That is,
+	\[ \phi(ab) = \phi(a)\phi(b),\ \forall a,b \in G \]
+	If there is an isomorphism from $G$ onto $\overline{G}$, we say that $G$ and $\overline{G}$ are \textit{isomorphic} and write $G \approx \overline{G}$.
+\end{definition}

part-2/chapters/chapter-6/properties-of-isomorphisms.tex

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+\section{Properties of Isomorphisms}
+
+\begin{theorem}[Properties of Isomorphisms Acting on Elements]
+	Suppose that $\phi$ is an isomorphism from a group $G$ onto a group $\overline{G}$. Then
+	\begin{enumerate}
+		\item $\phi$ carries the identity of $G$ to the identity of $\overline{G}$.
+		\item For every integer $n$ and for every group element $a$ in $G$, $\phi(a^n)=[\phi(a)]^n$.
+		\item For any elements $a$ and $b$ in $G$, $a$ and $b$ commute if and only if $\phi(a)$ and $\phi(b)$ commute.
+		\item $G = \lr{a}$ if and only if $\overline{G} = \lr{\phi(a)}$.
+		\item $\abs{a}=\abs{\phi(a)}$ for all $a$ in $G$ (isomorphisms preserve orders).
+		\item For a fixed integer $k$ and a fixed group element $b$ in $G$, the equation $x^k=b$ has the same number of solutions in $G$ as does the equation $x^k = \phi(b)$ in $\overline{G}$.
+		\item If $G$ is finite, then $G$ and $\overline{G}$ have exactly the same number of elements of every order.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[Properties of Isomorphisms Acting on Groups]
+	Suppose that $\phi$ is an isomorphism from a group $G$ onto a group $\overline{G}$. Then
+	\begin{enumerate}
+		\item $\phi^{-1}$ is an isomorphisms from $\overline{G}$ onto $G$.
+		\item $G$ is Abelian if and only if $\overline{G}$ is Abelian.
+		\item $G$ is cyclic if and only if $\overline{G}$ is cyclic.
+		\item If $K$ is a subgroup of $G$, then $\phi(K) = \{\phi(k)\ \vert\ k \in K\}$ is a subgroup of $\overline{G}$.
+		\item If $\overline{K}$ is a subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K}) = \{g \in G\ \vert\ \phi(g) \in \overline{K}\}$ is a subgroup of $G$.
+		\item $\phi(Z(G))=Z(\overline{G})$.
+	\end{enumerate}
+\end{theorem}