Created the Abstract Algebra theorems and definitions cheat sheet

part-2/chapters/chapter-7/an-application-of-cosets-to-permutation-groups.tex

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+\section{An Application of Cosets to Permutation Groups}
+
+\begin{definition}[Stabilizer of a Point]
+	Let $G$ be a group of permutations of a set $S$. For each $i$ in $S$, let stab$_G(i)=\{\phi \in G\ \vert\ \phi(i) = i\}$. We call stab$_G(i)$ the \textit{stabilizer of $i$ in $G$}.
+\end{definition}
+
+\begin{definition}[Orbit of a Point]
+	Let $G$ be a group of permutations of a set $S$. For each $s$ in $S$, let orb$_G(s)=\{\phi(s)\ \vert\ \phi \in G\}$. The set orb$_G(s)$ is a subset of $S$ called the \textit{orbit of $s$ under $G$}. We use $\abs{\text{orb}_G(s)}$ to denote the number of elements in orb$_G(s)$.
+\end{definition}
+
+\begin{theorem}[Orbit-Stabilizer Theorem]
+	Let $G$ be a finite group of permutations of a set $S$. Then, for any $i$ from $S$, $\abs{G} = \abs{\text{orb}_G(i)}\abs{\text{stab}_G(i)}$.
+\end{theorem}

part-2/chapters/chapter-7/chapter-7.tex

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+\chapter{Cosets and Lagrange's Theorem}
+\subimport{./}{properties-of-cosets.tex}
+\subimport{./}{lagranges-theorem-and-consequences.tex}
+\subimport{./}{an-application-of-cosets-to-permutation-groups.tex}
+\subimport{./}{the-rotation-group-of-a-cube-and-a-soccer-ball.tex}

part-2/chapters/chapter-7/lagranges-theorem-and-consequences.tex

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+\section{Lagrange's Theorem and Consequences}
+
+\begin{theorem}[Lagrange's Theorem: $\mathbf{\abs{H} \text{ Divides } \abs{G}}$]
+	If $G$ is a finite group and $H$ is a subgroup of $G$, then $\abs{H}$ divides $\abs{G}$. Moreover, the number of distinct left (right) cosets of $H$ in $G$ is $\abs{G}/\abs{H}$.
+\end{theorem}
+
+\begin{remark}
+	A special name and notation have been adopted for the number of left (or right) cosets of a subgroup in a group. The \textit{index} of a subgroup $H$ in $G$ is the number of distinct left cosets of $H$ in $G$. This number is denoted by $\abs{G:H}$.
+\end{remark}
+
+\begin{corollary}[$\mathbf{\abs{G:H} = \abs{G}/\abs{H}}$]
+	If $G$ is a finite group and $H$ is a subgroup of $G$, then $\abs{G:H} = \abs{G}/\abs{H}$.
+\end{corollary}
+
+\begin{corollary}[$\mathbf{\abs{a}}$ Divides $\mathbf{\abs{G}}$]
+	In a finite group, the order of each element of the group divides the order of the group.
+\end{corollary}
+
+\begin{corollary}[Groups of Prime Order Are Cyclic]
+	A group of prime order is cyclic.
+\end{corollary}
+
+\begin{corollary}[$\mathbf{a^{\abs{G}}=e}$]
+	Let $G$ be a finite group, and let $a \in G$. Then, $a^{\abs{G}} = e$.
+\end{corollary}
+
+\begin{corollary}[Fermat's Little Theorem]
+	For every integer $a$ and every prime $p$, $a^p \mod p = a \mod p$.
+\end{corollary}
+
+\begin{theorem}[$\mathbf{\abs{HK} = \abs{H}\abs{K}/\abs{H \cap K}}$]
+	For two finite subgroups $H$ and $K$ of a group, define the set $HK = \{hk\ \vert\ h \in H, k \in K\}$. Then $\abs{HK} = \abs{H}\abs{K}/\abs{H \cap K}$.
+\end{theorem}
+
+\begin{theorem}[Classification of Groups of order 2$\mathbf{p}$]
+	Let $G$ be a group of order $2p$, where $p$ is a prime greater than 2. Then $G$ is isomorphic to $\Z_{2p}$ or $D_p$.
+\end{theorem}

part-2/chapters/chapter-7/properties-of-cosets.tex

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+\section{Properties of Cosets}
+
+\begin{definition}[Coset of $\mathbf{H}$ in $\mathbf{G}$]
+	Let $G$ be a group and let $H$ be a nonempty subset of $G$. For any $a \in G$, the set $\{ah\ \vert\ h \in H\}$ is denoted by $aH$. Analogously, $Ha = \{ha\ \vert\ h \in H\}$ and $aHa^{-1} = \{aha^{-1}\ \vert\ h \in H\}$. When $H$ is a subgroup of $G$, the set $aH$ is called the \textit{left coset of $H$ in $G$ containing $a$}, whereas $Ha$ is called the \textit{right coset of $H$ in $G$ containing $a$}. In this case, the element $a$ is called the \textit{coset representative of $aH$ (or $Ha$)}. We use $\abs{aH}$ to denote the number of elements in the set $aH$, and $\abs{Ha}$ to denote the number of elements in $Ha$.
+\end{definition}
+
+\begin{lem}[ Properties of Cosets]
+	Let $H$ be a subgroup of $G$, and let $a$ and $b$ belong to $G$. Then,
+	\begin{enumerate}
+		\item $a \in aH$.
+		\item $aH = H$ if and only if $a \in H$.
+		\item $(ab)H = a(bH)$ and $H(ab) = (Ha)b$.
+		\item $aH = bH$ if and only if $a \in bH$.
+		\item $aH = bH$ or $aH \cap bH = \emptyset$.
+		\item $aH = bH$ if and only if $a^{-1}b \in H$.
+		\item $\abs{aH}=\abs{bH}$.
+		\item $aH = Ha$ if and only if $H = aHa^{-1}$.
+		\item $aH$ is a subgroup of $G$ if and only if $a \in H$.
+	\end{enumerate}
+\end{lem}

part-2/chapters/chapter-7/the-rotation-group-of-a-cube-and-a-soccer-ball.tex

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+\section{The Rotation Group of a Cube and a Soccer Ball}
+
+\begin{theorem}[The Rotation Group of a Cube]
+	The group of rotations of a cube is isomorphic to $S_4$.
+\end{theorem}