Created the Abstract Algebra theorems and definitions cheat sheet

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+\chapter{Group Homomorphisms}
+\subimport{./}{definition-and-examples.tex}
+\subimport{./}{properties-of-homomorphisms.tex}
+\subimport{./}{the-first-isomorphism-theorem.tex}
@@ -0,0 +1,9 @@
+\section{Definition and Examples}
+
+\begin{definition}[Group Homomorphism]
+	A \textit{homomorphism} $\phi$ from a group $G$ to a group $\overline{G}$ is a mapping from $G$ into $\overline{G}$ that preserves the group operation; that is, $\phi(ab) = \phi(a)\phi(b)$ for all $a, b$ in $G$.
+\end{definition}
+
+\begin{definition}[Kernel of a Homomorphism]
+	The \textit{kernel} of a homomorphism $\phi$ from a group $G$ to a group with identity $e$ is the set $\{x \in G\ \vert\ \phi(x)=e\}$. The kernel of $\phi$ is denoted by $\ker\phi$.
+\end{definition}
@@ -0,0 +1,32 @@
+\section{Properties of Homomorphisms}
+
+\begin{theorem}[Properties of Elements Under Homomorphisms]
+	Let $\phi$ be a homomorphism from a group $G$ to a group $\overline{G}$ and let $g$ be an element of $G$. Then
+	\begin{enumerate}
+		\item $\phi$ carries the identity of $G$ to $\overline{G}$.
+		\item $\phi(g^n)=(\phi(g))^n$ for all $n$ in $\Z$.
+		\item If $\abs{g}$ is finite, then $\abs{\phi(g)}$ divides $\abs{g}$.
+		\item $\ker\phi$ is a subgroup of $G$.
+		\item $\phi(a) = \phi(b)$ if and only if $a\ker\phi = b\ker\phi$.
+		\item If $\phi(g) = g'$, then $\phi^{-1}(g') = \{x \in G\ \vert\ \phi(x) = g'\} = g\ker\phi$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[Properties of Subgroups Under Homomorphisms]
+	Let $\phi$ be a homomorphism from a group $G$ to a group $\overline{G}$ and let $H$ be a subgroup of $G$. Then
+	\begin{enumerate}
+		\item $\phi(H) = \{\phi(h)\ \vert\ h \in H\}$ is a subgroup of $\overline{G}$.
+		\item If $H$ is cyclic, then $\phi(H)$ is cyclic.
+		\item If $H$ is Abelian, then $\phi(H)$ is Abelian.
+		\item If $H$ is normal in $G$, then $\phi(H)$ is normal in $\phi(G)$.
+		\item If $\abs{\ker\phi} = n$, then $\phi$ is an $n$-to-1 mapping from $G$ onto $\phi(G)$.
+		\item If $\abs{H} = n$, then $\abs{\phi(H)}$ divides $n$.
+		\item If $\overline{K}$ is a subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K})=\{k \in G\ \vert\ \phi(k) \in \overline{K}\}$ is a subgroup of $G$.
+		\item If $\overline{K}$ is a normal subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K})=\{ k \in G\ \vert\ \phi(k) \in \overline{K}\}$ is a normal subgroup of $G$.
+		\item If $\phi$ is onto and $\ker\phi = \{e\}$, then $\phi$ is an isomorphism from $G$ to $\overline{G}$.
+	\end{enumerate}
+\end{theorem}
+
+\begin{corollary}[Kernels Are Normal]
+	Let $\phi$ be a group homomorphism from $G$ to $\overline{G}$. Then $\ker\phi$ is a normal subgroup of $G$.
+\end{corollary}
@@ -0,0 +1,13 @@
+\section{The First Isomorphism Theorem}
+
+\begin{theorem}[First Isomorphism Theorem (Jordan, 1870)]
+	Let $\phi$ be a group homomorphism from $G$ to $\overline{G}$. Then the mapping from $G/\ker\phi$ to $\phi(G)$, given by $g\ker\phi \to \phi(g)$, is an isomorphism. In symbols, $G/\ker\phi \approx \phi(G)$.
+\end{theorem}
+
+\begin{corollary}
+	If $\phi$ is a homomorphism from a finite group $G$ to $\overline{G}$, then $\abs{\phi(G)}$ divides $\abs{G}$ and $\abs{\overline{G}}$.
+\end{corollary}
+
+\begin{theorem}[Normal Subgroups Are Kernels]
+	Every normal subgroup of a group $G$ is the kernel of a homomorphism of $G$. In particular, a normal subgroup $N$ is the kernel of the mapping $g \to gN$ from $G$ to $G/N$.
+\end{theorem}
@@ -0,0 +1,4 @@
+\chapter{Fundamental Theorem of Finite Abelian Groups}
+\subimport{./}{the-fundamental-theorem.tex}
+\subimport{./}{the-isomorphism-classes-of-abelian-groups.tex}
+\subimport{./}{proof-of-the-fundamental-theorem.tex}
@@ -0,0 +1,17 @@
+\section{Proof of the Fundamental Theorem}
+
+\begin{lemma}
+	Let $G$ be a finite Abelian group of order $p^nm$, where $p$ is a prime that does not divide $m$. Then $G = H \times K$, where $H = \{x \in G\ \vert\ x^{p^n} =e\}$ and $K =\{x \in G\ \vert\ x^m = e\}$. Moreover, $\abs{H}=p^n$.
+\end{lemma}
+
+\begin{lemma}
+	Let $G$ be an Abelian group of prime-power order and let $a$ be an element of maximum order in $G$. Then $G$ can be written in the form $\lr{a} \times K$.
+\end{lemma}
+
+\begin{lemma}
+	A finite Abelian group of prime-power order is an internal direct product of cyclic groups.
+\end{lemma}
+
+\begin{lemma}
+	Suppose that $G$ is a finite Abelian group of prime-power order. If $G=H_1 \times H_2 \times \dots \times H_m$ and $G=K_1 \times K_2 \times \dots \times K_n$, where the $H$'s and $K$'s are nontrivial cyclic subgroups with $\abs{H_1} \geq \abs{H_2} \geq \dots \geq \abs{H_m}$ and $\abs{K_1} \geq \abs{K_2} \geq \dots \geq \abs{K_n}$, then $m=n$ and $\abs{H_i} = \abs{K_i}$ for all $i$.
+\end{lemma}
@@ -0,0 +1,5 @@
+\section{The Fundamental Theorem}
+
+\begin{theorem}[Fundamental Theorem of Finite Abelian Groups]
+	Every finite Abelian group is a direct product of cyclic groups of prime-power order. Moreover, the number of terms in the product and the orders of the cyclic groups are uniquely determined by the group.
+\end{theorem}
@@ -0,0 +1,16 @@
+\section{The Isomorphism Classes of Abelian Groups}
+
+\begin{remark}[Greedy Algorithm for an Abelian Group of Order $\mathbf{p^n}$]
+	The Fundamental Theorem is extremely powerful. As an application, we can use it as an algorithm for constructing all Abelian groups of any order. Let's look at Abelian groups of a certain order $n$, where $n$ has two or more distinct prime divisors.
+	\begin{enumerate}
+		\item Compute the orders of the elements of the group $G$
+		\item Select an element $a_1$ of maximum order and define $G_1 = \lr{a_1}$. Set $i = 1$.
+		\item If $\abs{G} = \abs{G_i}$, stop. Otherwise, replace $i$ by $i + 1$.
+		\item Select an element $a_i$ of maximum order $p^k$ such that $p^k \leq \abs{G}/\abs{G_{i-1}}$ and none of $a_i, a^p_i,a^{p^2}_i, \dots, a^{p^{k-1}}_i$ is in $G_{i-1}$, and define $G_i=G_{i-1} \times \lr{a_i}$.
+		\item Return to step 3.
+	\end{enumerate}
+\end{remark}
+
+\begin{corollary}[Existence of Subgroups of Abelian Groups]
+	If $m$ divides the order of a finite Abelian group $G$, then $G$ has a subgroup of order $m$.
+\end{corollary}
@@ -0,0 +1,3 @@
+\chapter{Groups}
+\subimport{./}{definition-and-examples-of-groups.tex}
+\subimport{./}{elementary-properties-of-groups.tex}
@@ -0,0 +1,14 @@
+\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}
@@ -0,0 +1,17 @@
+\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}
@@ -0,0 +1,3 @@
+\chapter{Finite Groups; Subgroups}
+\subimport{./}{subgroup-tests.tex}
+\subimport{./}{terminology-and-notation.tex}
@@ -0,0 +1,13 @@
+\section{Terminology and Notation}
+
+\begin{definition}[Order of a Group]
+	The number of elements of a group (finite or infinite) is called its \textit{order}. We will use $\abs{G}$ to denote the order of $G$.
+\end{definition}
+
+\begin{definition}[Order of an Element]
+	The \textit{order} of an element $g$ in a group $G$ is the smallest positive integer $n$ such that $g^n = e$. (In additive notation, this would be $ng = 0$.) If no such integer exists, we say that $g$ has \textit{infinite order}. The order of an element $g$ is denoted by $\abs{g}$.
+\end{definition}
+
+\begin{definition}[Subgroup]
+	If a subset $H$ of a group $G$ is itself a group under the operation of $G$, we say that $H$ is a \textit{subgroup} of $G$.
+\end{definition}
@@ -0,0 +1,36 @@
+\section{Subgroup Tests}
+
+\begin{theorem}[One-Step Subgroup Test]
+	Let $G$ be a group and $H$ a nonempty subset of $G$. If $ab^{-1}$ is in $H$ whenever $a$ and $b$ are in $H$, then $H$ is a subgroup of $G$. (In additive notation, if $a - b$ is in $H$ whenever $a$ and $b$ are in $H$, then $H$ is a subgroup of $G$.)
+\end{theorem}
+
+\begin{theorem}[Two-Step Subgroup Test]
+	Let $G$ be a group and let $H$ be a nonempty subset of $G$. If $ab$ is in $H$ whenever $a$ and $b$ are in $H$ ($H$ is closed under the operation), and $a^{-1}$ is in $H$ whenever $a$ is in $H$ ($H$ is closed under taking inverses), then $H$ is a subgroup of $G$.
+\end{theorem}
+
+\begin{theorem}[Finite Subgroup Test]
+	Let $H$ be a nonempty finite subset of a group $G$. If $H$ is closed under the operation of $G$, then $H$ is a subgroup of $G$.
+\end{theorem}
+
+\begin{theorem}[$\mathbf{\lr{a}}$ Is a Subgroup]
+	Let $G$ be a group, and let $a$ be any element of $G$. Then, $\lr{a}$ is a subgroup of $G$.
+\end{theorem}
+
+\begin{definition}[Center of a Group]
+	The \textit{center}, $Z(G)$, of a group $G$ is the subset of elements in $G$ that commute with every element of $G$. In symbols,
+	\[ Z(G) = \{a \in G\ \vert\ ax = xa,\ \forall\ x \in G\} \]
+	[The notation $Z(G)$ comes from the fact that the German word for center is \textit{Zentrum}. The term was coined by J.A. de Séguier in 1904.]
+\end{definition}
+
+\begin{theorem}[Center Is a Subgroup]
+	The center of a group $G$ is a subgroup of $G$.
+\end{theorem}
+
+\begin{definition}[Centralizer of $\mathbf{a}$ in $\mathbf{G}$]
+	Let $a$ be a fixed element of a group $G$. The \textit{centralizer of $a$ in $G$}, $C(a)$, is the set of all elements in $G$ that commute with $a$. In symbols,
+	\[ C(a) = \{g \in G\ \vert\ ga = ag\} \]
+\end{definition}
+
+\begin{theorem}[$\mathbf{C(a)}$ Is a Subgroup]
+	For each $a$ in a group $G$, the centralizer of $a$ is a subgroup of $G$.
+\end{theorem}
@@ -0,0 +1,3 @@
+\chapter{Cyclic Groups}
+\subimport{./}{properties-of-cyclic-groups.tex}
+\subimport{./}{classification-of-subgroups-of-cyclic-groups.tex}
@@ -0,0 +1,17 @@
+\section{Classification of Subgroups of Cyclic Groups}
+
+\begin{theorem}[Fundamental Theorem of Cyclic Groups]
+	Every subgroup of a cyclic group is cyclic. Moreover, if $\abs{\lr{a}} = n$, then the order of any subgroup of $\lr{a}$ is a divisor of $n$; and, for each, positive divisor $k$ of $n$, the group $\lr{a}$ has exactly one subgroup of order $k$ -- namely, $\lr{a^{n/k}}$.
+\end{theorem}
+
+\begin{corollary}[Subgroups of $\mathbf{\Z_n}$]
+	For each positive divisor $k$ of $n$, the set $\lr{n/k}$ is the unique subgroup of $\Z_n$ of order $k$; moreover, these are the only subgroups of $\Z_n$.
+\end{corollary}
+
+\begin{theorem}[Number of Elements of Each Order in a Cyclic Group]
+	If $d$ is a positive divisor of $n$, the number of elements of order $d$ in a cyclic group of order $n$ is $\phi(d)$.
+\end{theorem}
+
+\begin{corollary}[Number of Elements of Order $\mathbf{d}$ in a Finite Group]
+	In a finite group, the number of elements of order $d$ is a multiple of $\phi(d)$.
+\end{corollary}
@@ -0,0 +1,33 @@
+\section{Properties of Cyclic Groups}
+
+\begin{theorem}[Criterion for $\mathbf{a^i=a^j}$]
+	Let $G$ be a group, and let $a$ belong to $G$. If $a$ has infinite order, then $a^i = a^j$ if and only if $i = j$. If $a$ has finite order, say, $n$, then $\lr{a} = \{e, a, a^2, \dots, a^{n - 1}\}$ and $a^i = a^j$ if and only if $n$ divides $i - j$.
+\end{theorem}
+
+\begin{corollary}[$\mathbf{\abs{a}=\abs{\lr{a}}}$]
+	For any group element $a$, $\abs{a} = \abs{\lr{a}}$.
+\end{corollary}
+
+\begin{corollary}[$\mathbf{a^k = e}$ Implies That $\mathbf{\abs{a}}$ Divides $\mathbf{k}$]
+	Let $G$ be a group and let $a$ be an element of order $n$ in $G$. If $a^k = e$, then $n$ divides $k$.
+\end{corollary}
+
+\begin{theorem}[$\mathbf{\lr{a^k} = \lr{a^{\textbf{gcd}(n,k)}}}$ and $\mathbf{\abs{a^k} = n/\textbf{gcd}(n,k)}$]
+	Let $a$ be an element of order $n$ in a gruop and let $k$ be a positive integer. Then $\lr{a^k} = \lr{a^{\gcd(n,k)}}$ and $\abs{a^k} = n / \gcd(n,k)$.
+\end{theorem}
+
+\begin{corollary}[Orders of Elements in Finite Cyclic Groups]
+	In a finite cyclic group, the order of an element divides the order of the group.
+\end{corollary}
+
+\begin{corollary}[Criterion for $\mathbf{\lr{a^i} = \lr{a^j}}$ and $\mathbf{\abs{a^i} = \abs{a^j}}$]
+	Let $\abs{a} = n$. Then $\lr{a^i} = \lr{a^j}$ if and only if $\gcd(n, i) = \gcd(n,j)$, and $\abs{a^i} = \abs{a^j}$ if and only if $\gcd(n,i) = \gcd(n,j)$.
+\end{corollary}
+
+\begin{corollary}[Generators of Finite Cyclic Groups]
+	Let $\abs{a} = n$. Then $\lr{a} = \lr{a^j}$ if and only if $\gcd(n,j) = 1$, and $\abs{a} = \abs{\lr{a^j}}$ if and only if $\gcd(n,j) = 1$.
+\end{corollary}
+
+\begin{corollary}[Generators of $\mathbf{\Z_n}$]
+	An integer $k$ in $\Z_n$ is a generator of $Z_n$ if and only if $\gcd(n,k) = 1$.
+\end{corollary}
@@ -0,0 +1,4 @@
+\chapter{Permutation Groups}
+\subimport{./}{definition-and-notation.tex}
+\subimport{./}{cycle-notation.tex}
+\subimport{./}{properties-of-permutations.tex}
@@ -0,0 +1,28 @@
+\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}
@@ -0,0 +1,5 @@
+\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}
@@ -0,0 +1,43 @@
+\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}
@@ -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}
@@ -0,0 +1,5 @@
+\section{Cayley's Theorem}
+
+\begin{theorem}[Cayley's Theorem (1854)]
+	Every group is isomorphic to a group of permutations.
+\end{theorem}
@@ -0,0 +1,5 @@
+\chapter{Isomorphisms}
+\subimport{./}{definition-and-examples.tex}
+\subimport{./}{cayleys-theorem.tex}
+\subimport{./}{properties-of-isomorphisms.tex}
+\subimport{./}{automorphisms.tex}
@@ -0,0 +1,7 @@
+\section{Definition and Examples}
+
+\begin{definition}[Group Isomorphism]
+	An \textit{isomorphism} $\phi$ from a group $G$ to a group $\overline{G}$ is a one-to-one mapping (or function) from $G$ onto $\overline{G}$ that preserves the group operation. That is,
+	\[ \phi(ab) = \phi(a)\phi(b),\ \forall a,b \in G \]
+	If there is an isomorphism from $G$ onto $\overline{G}$, we say that $G$ and $\overline{G}$ are \textit{isomorphic} and write $G \approx \overline{G}$.
+\end{definition}
@@ -0,0 +1,26 @@
+\section{Properties of Isomorphisms}
+
+\begin{theorem}[Properties of Isomorphisms Acting on Elements]
+	Suppose that $\phi$ is an isomorphism from a group $G$ onto a group $\overline{G}$. Then
+	\begin{enumerate}
+		\item $\phi$ carries the identity of $G$ to the identity of $\overline{G}$.
+		\item For every integer $n$ and for every group element $a$ in $G$, $\phi(a^n)=[\phi(a)]^n$.
+		\item For any elements $a$ and $b$ in $G$, $a$ and $b$ commute if and only if $\phi(a)$ and $\phi(b)$ commute.
+		\item $G = \lr{a}$ if and only if $\overline{G} = \lr{\phi(a)}$.
+		\item $\abs{a}=\abs{\phi(a)}$ for all $a$ in $G$ (isomorphisms preserve orders).
+		\item For a fixed integer $k$ and a fixed group element $b$ in $G$, the equation $x^k=b$ has the same number of solutions in $G$ as does the equation $x^k = \phi(b)$ in $\overline{G}$.
+		\item If $G$ is finite, then $G$ and $\overline{G}$ have exactly the same number of elements of every order.
+	\end{enumerate}
+\end{theorem}
+
+\begin{theorem}[Properties of Isomorphisms Acting on Groups]
+	Suppose that $\phi$ is an isomorphism from a group $G$ onto a group $\overline{G}$. Then
+	\begin{enumerate}
+		\item $\phi^{-1}$ is an isomorphisms from $\overline{G}$ onto $G$.
+		\item $G$ is Abelian if and only if $\overline{G}$ is Abelian.
+		\item $G$ is cyclic if and only if $\overline{G}$ is cyclic.
+		\item If $K$ is a subgroup of $G$, then $\phi(K) = \{\phi(k)\ \vert\ k \in K\}$ is a subgroup of $\overline{G}$.
+		\item If $\overline{K}$ is a subgroup of $\overline{G}$, then $\phi^{-1}(\overline{K}) = \{g \in G\ \vert\ \phi(g) \in \overline{K}\}$ is a subgroup of $G$.
+		\item $\phi(Z(G))=Z(\overline{G})$.
+	\end{enumerate}
+\end{theorem}
@@ -0,0 +1,13 @@
+\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}
@@ -0,0 +1,5 @@
+\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}
@@ -0,0 +1,37 @@
+\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}
@@ -0,0 +1,20 @@
+\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}
@@ -0,0 +1,5 @@
+\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}
@@ -0,0 +1,4 @@
+\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}
@@ -0,0 +1,5 @@
+\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}
@@ -0,0 +1,18 @@
+\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}
@@ -0,0 +1,17 @@
+\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}
@@ -0,0 +1,13 @@
+\section{Applications of Factor Groups}
+
+\begin{theorem}[$\mathbf{G/Z}$ Theorem]
+	Let $G$ be a group and let $Z(G)$ be the center of $G$. If $G/Z(G)$ is cyclic, then $G$ is Abelian.
+\end{theorem}
+
+\begin{theorem}[$\mathbf{G/Z(G) \approx \text{Inn}(G)}$]
+	For any group $G$, $G/Z(G)$ is isomorphic to Inn$(G)$.
+\end{theorem}
+
+\begin{theorem}[Cauchy's Theorem for Abelian Groups]
+	Let $G$ be a finite Abelian group and let $p$ be a prime that divides the order of $G$. Then $G$ has an element of order $p$.
+\end{theorem}
@@ -0,0 +1,5 @@
+\chapter{Normal Subgroups and Factor Groups}
+\subimport{./}{normal-subgroups.tex}
+\subimport{./}{factor-groups.tex}
+\subimport{./}{applications-of-factor-groups.tex}
+\subimport{./}{internal-direct-products.tex}
@@ -0,0 +1,5 @@
+\section{Factor Groups}
+
+\begin{theorem}[Factor Groups (O. Hölder, 1889)]
+	Let $G$ be a group and let $H$ be a normal subgroup of $G$. The set $G/H = \{ aH\ \vert\ a \in G\}$ is a group under the operation $(aH)(bH) = abH$.
+\end{theorem}
@@ -0,0 +1,26 @@
+\section{Internal Direct Products}
+
+\begin{definition}[Internal Direct Product of $\mathbf{H}$ and $\mathbf{K}$]
+	We say that $G$ is the \textit{internal direct product} of $H$ and $K$ and write $G = H \times K$ if $H$ and $K$ are normal subgroups of $G$ and
+	\[ G = HK\ \ \ \ \text{and}\ \ \ \ H \cap K = \{e\} \]
+\end{definition}
+
+\begin{definition}[Internal Direct Product $\mathbf{H_1 \times H_2 \times \dots \times H_n}$]
+	Let $H_1, H_2,\dots,H_n$ be a finite collection of normal subgroups of $G$. We say that $G$ is the \textit{internal direct product} of $H_1,H_2,\dots,H_n$ and write $G=H_1\times H_2 \times \dots \times H_n$, if
+	\begin{enumerate}
+		\item $G = H_1H_2\dots H_n = \{h_1h_2\dots h_n\ \vert\ h_i \in H_i\}$,
+		\item $(H_1H_2\dots H_n) \cap H_{i + 1} = {e}$ for $i=1,2,\dots, n-1$.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}[$\mathbf{H_1 \times H_2 \times \dots \times H_n \approx H_1 \oplus H_2 \oplus \dots \oplus H_n}$]
+	If a group $G$ is the internal direct product of a finite number of subgroups $H_1,H_2, \dots, H_n$, then $G$ is isomorphic to the external direct product of $H_1,H_2 \dots, H_n$.
+\end{theorem}
+
+\begin{theorem}[Classification of Groups of Order $\mathbf{p^2}$]
+	Every group of order $p^2$, where $p$ is a prime, is isomorphic to $\Z_{p^2}$ or $\Z_p \oplus \Z_p$.
+\end{theorem}
+
+\begin{corollary}
+	If $G$ is a group of order $p^2$, where $p$ is a prime, then $G$ is Abelian.
+\end{corollary}
@@ -0,0 +1,9 @@
+\section{Normal Subgroups}
+
+\begin{definition}[Normal Subgroup]
+	A subgroup $H$ of a group $G$ is called a \textit{normal} subgroup of $G$ if $aH = Ha$ for all $a$ in $G$. We denote this by $H \triangleleft G$.
+\end{definition}
+
+\begin{theorem}[Normal Subgroup Test]
+	A subgroup $H$ of $G$ is normal in $G$ if and only if $xHx^{-1} \subseteq H$ for all $x$ in $G$.
+\end{theorem}