Created the Abstract Algebra theorems and definitions cheat sheet

part-2/chapters/chapter-8/chapter-8.tex

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+\chapter{External Direct Products}
+\subimport{./}{definition-and-examples.tex}
+\subimport{./}{properties-of-external-direct-products.tex}
+\subimport{./}{the-group-of-units-modulo-n-as-an-external-direct-product.tex}

part-2/chapters/chapter-8/definition-and-examples.tex

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+\section{Definition and Examples}
+
+\begin{definition}[External Direct Product]
+	Let $G_1,G_2,\dots,G_n$ be a finite collection of groups. The \textit{external direct product} of $G_1,G_2,\dots,G_n$, written as $G_1 \oplus G_2 \oplus \dots \oplus G_n$, is the set of all $n$-tuples for which the $i$th component is an element of $G_i$ and the operation is componentwise.
+\end{definition}

part-2/chapters/chapter-8/properties-of-external-direct-products.tex

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+\section{Properties of External Direct Products}
+
+\begin{theorem}[Order of an Element in a Direct Product]
+	The order of an element in a direct product of a finite number of finite groups is the least common multiple of the orders of the component of the element. In symbols,
+	\[ \abs{(g_1,g_2,\dots,g_n)} = \lcm(\abs{g_1},\abs{g_2},\dots,\abs{g_n}) \]
+\end{theorem}
+
+\begin{theorem}[Criterion for $\mathbf{G \oplus H}$ to be Cyclic]
+	Let $G$ and $H$ be finite cyclic groups. Then $G \oplus H$ is cyclic if and only if $\abs{G}$ and $\abs{H}$ are relatively prime.
+\end{theorem}
+
+\begin{corollary}[Criterion for $\mathbf{G_1 \oplus G_2 \oplus \dots \oplus G_n}$ to Be Cyclic]
+	An external direct product $G_1 \oplus G_2 \oplus \dots \oplus G_n$ of a finite number of finite cyclic groups is cyclic if and only if $\abs{G_i}$ and $\abs{G_j}$ are relatively prime when $i \neq j$.
+\end{corollary}
+
+\begin{corollary}[Criterion for $\mathbf{\Z_{n_1n_2\dots n_k} \approx \Z_{n_1} \oplus \Z_{n_2} \oplus \dots \oplus \Z_{n_k}}$]
+	Let $m = n_1n_2\dots n_k$. Then $\Z_m$ is isomorphic to $\Z_{n_1} \oplus \Z_{n_2} \oplus \dots \oplus \Z_{n_k}$ if and only if $n_i$ and $n_j$ are relatively prime when $i \neq j$.
+\end{corollary}

part-2/chapters/chapter-8/the-group-of-units-modulo-n-as-an-external-direct-product.tex

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+\section{The Group of Units Modulo $\mathbf{n}$ as an External Direct Product}
+
+\begin{remark}
+	The $U$-groups provide a convenient way to illustrate the preceding ideas. We first introduce some notation. If $k$ is a divisor of $n$, let
+	\[ U_k(n) = \{x \in U(n)\ \vert\ x \mod k = 1\} \]
+\end{remark}
+
+\begin{theorem}[$\mathbf{U(n)}$ as an External Direct Product]
+	Suppose $s$ and $t$ are relatively prime. Then $U(st)$ is isomorphic to the external direct product of $U(s)$ and $U(t)$. In short,
+	\[ U(st) \approx U(s) \oplus U(t) \]
+	Moreover, $U_s(st)$ is isomorphic to $U(t)$ and $U_t(st)$ is isomorphic to $U(s)$.
+\end{theorem}
+
+\begin{corollary}
+	Let $m = n_1n_2\dots n_k$, where $\gcd(n_i,n_j)=1$ for $i \neq j$. Then,
+	\[ U(m) \approx U(n_1) \oplus U(n_2) \oplus \dots \oplus U(n_k) \]
+\end{corollary}