Created the Abstract Algebra theorems and definitions cheat sheet

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

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+\chapter{Groups}
+\subimport{./}{definition-and-examples-of-groups.tex}
+\subimport{./}{elementary-properties-of-groups.tex}

part-2/chapters/chapter-2/definition-and-examples-of-groups.tex

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+\section{Definition and Examples of Groups}
+
+\begin{definition}[Binary Operation]
+	Let $G$ be a set. A \textit{binary operation} on $G$ is a function that assigns each ordered pair of elements of $G$ an element of $G$.
+\end{definition}
+
+\begin{definition}[Group]
+	Let $G$ be a set together with a binary operation (usually called multiplication) that assigns to each ordered pair $(a, b)$ of elements of $G$ an element in $G$ denoted by $ab$. We say $G$ is a \textit{group} under this operation if the following three properties are satisfied.
+	\begin{enumerate}
+		\item \textit{Associativity}. The operation is associative; that is, $(ab)c = a(bc)$ for all $a,b,c$ in $G$.
+		\item \textit{Identity}. There is an element $e$ (called the \textit{identity}) in $G$ such that $ae = ea = a$ for all $a$ in $G$.
+		\item \textit{Inverses}. For each element $a$ in $G$, there is an element $b$ in $G$ (called an \textit{inverse} of $a$) such that $ab = ba = e$.
+	\end{enumerate}
+\end{definition}

part-2/chapters/chapter-2/elementary-properties-of-groups.tex

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+\section{Elementary Properties of Groups}
+
+\begin{theorem}[Uniqueness of the Identity]
+	In a group $G$, there is only one identity element.
+\end{theorem}
+
+\begin{theorem}[Cancellation]
+	In a group $G$, the right and left cancellation laws hold; that is, $ba = ca$ implies $b = c$, and $ab = ac$ implies $b = c$.
+\end{theorem}
+
+\begin{theorem}[Uniqueness of Inverses]
+	For each element $a$ in a group $G$, there is a unique element $b$ in $G$ such that $ab = ba = e$.
+\end{theorem}
+
+\begin{theorem}[Socks-Shoes Property]
+	For group elements $a$ and $b$, $(ab)^{-1} = b^{-1}a^{-1}$.
+\end{theorem}