Created the Abstract Algebra theorems and definitions cheat sheet

part-2/chapters/chapter-5/chapter-5.tex

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+\chapter{Permutation Groups}
+\subimport{./}{definition-and-notation.tex}
+\subimport{./}{cycle-notation.tex}
+\subimport{./}{properties-of-permutations.tex}

part-2/chapters/chapter-5/cycle-notation.tex

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+\section{Cycle Notation}
+\begin{definition}
+	Consider the permutation
+	\[ \alpha = \begin{bmatrix}
+			1 & 2 & 3 & 4 & 5 & 6 \\
+			2 & 1 & 4 & 6 & 5 & 3
+		\end{bmatrix}\]
+
+	The assignment of values is as follows:
+
+	\begin{align*}
+		1 & \mapsto 2 \\
+		2 & \mapsto 1 \\
+		3 & \mapsto 4 \\
+		4 & \mapsto 6 \\
+		5 & \mapsto 5 \\
+		6 & \mapsto 3
+	\end{align*}
+
+	Although mathematically satisfactory, such diagrams are cumbersome. Instead, we leave out the arrows and simply write $\alpha = (1,2)(3,4,6)(5)$.
+
+	It is also worth noting that an expression of the form $(a_1, a_2, \dots, a_m)$ is called a \textit{cycle of length $m$}, or an \textit{$m$-cycle}.
+\end{definition}
+\begin{example}
+	To multiply cycles, consider the following permutations from $S_8$. Let $\alpha = (13)(27)(456)(8)$ and $\beta = (1237)(648)(5)$. (When the domain consists of single-digit integers, it is common practice to omit the commas between the digits.) What is the cycle form of $\alpha\beta$? Of course, one could say that $\alpha\beta = (13)(27)(456)(8)(1237)(648)(5)$, but it is usually more desirable to express a permutation in a \textit{disjoint} cycle form (that is, the various cycles have no number in common). Well, keeping in mind that function composition is done from right to left and that each cycle that does not contain a symbol fixes the symbol, we observe that $(5)$ fixes 1; $(648)$ fixes $1$; $(1237)$ sends 1 to 2, $(8)$ fixes 2; $(456)$ fixes 2; $(27)$ sends 2 to 7; and $(13)$ fixes 7. So the net effect of $\alpha\beta$ is to send 1 to 7. Thus, we begin $\alpha\beta=(17\dots)\dots$. Now, repeating the entire process beginning with 7, we have, cycle by cycle, right to left,
+	\[ 7 \to 7 \to 7 \to 1 \to 1 \to 1 \to 1 \to 3, \]
+	so that $\alpha\beta = (173\dots)\dots$. Ultimately, we have $\alpha\beta = (1732)(48)(56)$. The import thing to bear in mind when multiplying cycles is to "keep moving" from one cycle to the next from right to left.
+\end{example}

part-2/chapters/chapter-5/definition-and-notation.tex

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+\section{Definition and Notation}
+
+\begin{definition}[Permutation of $\mathbf{A}$, Permutation Group of $\mathbf{A}$]
+	A \textit{permutation} of a set $A$ is a function from $A$ to $A$ that is both one-to-one and onto. A \textit{permutation group} of a set $A$ is a set of permutations of $A$ that forms a group under function composition.
+\end{definition}

part-2/chapters/chapter-5/properties-of-permutations.tex

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+\section{Properties of Permutations}
+
+\begin{theorem}[Products of Disjoint Cycles]
+	Every permutation of a finite set can be written as a cycle or as a product of disjoint cycles.
+\end{theorem}
+
+\begin{theorem}[Disjoint Cycles Commute]
+	If the pair of cycles $\alpha = (a_1, a_2, \dots, a_m)$ and $\beta = (b_1, b_2, \dots, b_n)$ have no entries in common, then $\alpha\beta = \beta\alpha$.
+\end{theorem}
+
+\begin{theorem}[Order of a Permutation (Ruffini, 1799)]
+	The order of a permutation of a finite set written in disjoint cycle form is the least common multiple of the lengths of the cycles.
+\end{theorem}
+
+\begin{theorem}[Product of 2-Cycles]
+	Every permutation in $S_n,\ n>1$ is a product of 2-cycles.
+\end{theorem}
+
+\begin{lem}
+	If $\varepsilon = \beta_1\beta_2\dots\beta_r$, where the $\beta$'s are 2-cycles, then $r$ is even.
+\end{lem}
+
+\begin{theorem}[Always Even or Always Odd]
+	If a permutation $\alpha$ can be expressed as a product of an even (odd) number of 2-cycles, then every decomposition of $\alpha$ into a product of 2-cycles must have an even (odd) number of 2-cycles. In symbols, if
+	\[ \alpha = \beta_1\beta_2\dots\beta_r\ \ \ \ \text{and}\ \ \ \ \alpha=\gamma_1\gamma_2\dots\gamma_s, \]
+	where the $\beta$'s and the $\gamma$'s are 2-cycles, then $r$ and $s$ are both even or both odd.
+\end{theorem}
+
+\begin{definition}[Even and Odd Permutations]
+	A permutation that can be expressed as a product of an even number of 2-cycles is called an \textit{even} permutation. A permutation that can be expressed as a product of an odd number of 2-cycles is called an \textit{odd} permutation.
+\end{definition}
+
+\begin{theorem}[Even Permutations Form a Group]
+	The set of even permutations in $S_n$ forms a subgroup of $S_n$.
+\end{theorem}
+
+\begin{definition}[Alternating Group of Degree $\mathbf{n}$]
+	The group of even permutations of $n$ symbols is denoted by $A_n$ and is called the \textit{alternating group of degree $n$}.
+\end{definition}
+
+\begin{theorem}
+	For $n > 1$, $A_n$ has order $n!/2$.
+\end{theorem}