Created the Abstract Algebra theorems and definitions cheat sheet

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

@@ -0,0 +1,19 @@
+\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}