Added group theory parts

Advanced/Group Theory/main.tex

@@ -13,12 +13,13 @@
 		<Fall 2022>
 		{Group Theory}
 		{
-			Based on a lesson by Janet Chen \\
 			Prepared by Mark on \today
 		}
 
 
 	\input{parts/00 review}
 	\input{parts/01 groups}
+	\input{parts/02 isomorphism}
+	\input{parts/03 bonus}
 
 \end{document}

Advanced/Group Theory/parts/01 groups.tex

@@ -7,8 +7,8 @@ A group must have the following properties: \\
 \begin{enumerate}
 	\item $G$ is closed under $\ast$. In other words, $a, b \in G \implies a \ast b \in G$.
 	\item $\ast$ is associative: $(a \ast b) \ast c = a \ast (b \ast c)$
-	\item There is an \textit{identity} $\overline{0} \in G$, so that $a \ast \overline{0} = a$ for all $a \in G$.
-	\item For any $a \in G$, there exists a $b \in G$ so that $a \ast b = \overline{0}$. $b$ is called the \textit{inverse} of $a$. \\
+	\item There is an \textit{identity} $e \in G$, so that $a \ast e = a \ast e = a$ for all $a \in G$.
+	\item For any $a \in G$, there exists a $b \in G$ so that $a \ast b = b \ast a = e$. $b$ is called the \textit{inverse} of $a$. \\
 	This element is written as $-a$ if our operator is addition and $a^{-1}$ otherwise.
 \end{enumerate}
 
@@ -59,6 +59,11 @@ Is $(G, \circ)$ a group?
 \vfill
 \pagebreak
 
+\problem{}
+Show that if $G$ has four elements, $(G, \ast)$ is abelian.
+
+\vfill
+
 \problem{}
 Show that a group has exactly one identity element.
 \vfill
@@ -67,6 +72,18 @@ Show that a group has exactly one identity element.
 Show that each element in a group has exactly one inverse.
 \vfill
 
+\problem{}
+Show that...
+\begin{itemize}
+	\item $e^{-1} = 1$
+	\item $(a^{-1})^{-1} = a$
+\end{itemize}
+\vfill
+
+\problem{}
+Show that $(a^m)^{-1} = (a^{-1})^m$ for all $a \in G$ and $m \in \mathbb{Z}$.
+\vfill
+
 \problem{}
 Let $(G, \ast)$ be a group and $a, b, c \in G$. Show that...
 \begin{itemize}
@@ -75,15 +92,16 @@ Let $(G, \ast)$ be a group and $a, b, c \in G$. Show that...
 \end{itemize}
 What does this mean intuitively?
 \vfill
+\pagebreak
 
 \problem{}
 Let $(G, \ast)$ be a finite group (i.e, $G$ has finitely many elements), and let $g \in G$. \\
-Show that $\exists~n \in Z^+$ so that $g^n = \overline{0}$ \\
+Show that $\exists~n \in Z^+$ so that $g^n = e$ \\
 \hint{$g^n = g \ast g \ast ... \ast g$ $n$ times.}
 
 \vspace{2mm}
 
-The smallest such $n$ defines the \textit{order} of $(G, \ast)$.
+The smallest such $n$ defines the \textit{order} of $g$.
 
 \vfill
 
@@ -92,42 +110,48 @@ What is the order of 5 in $(\mathbb{Z}/25, +)$? \\
 What is the order of 2 in $((\mathbb{Z}/17)^\times, \times)$? \\
 
 \vfill
+\pagebreak
 
 \problem{}
-Show that if $G$ has four elements, $(G, \ast)$ is abelian.
+Let $e, a, b, c$ be counterclockwise rotations of a square by $0, \frac{\pi}{2}, \pi,$ and $\frac{3\pi}{2}$. \\
+Create a multiplication table for this group.
+\vfill
+
+\problem{}
+Let $d, f, g, h$ correspond to reflections of the square along the following axis. \\
+Create a multiplication table for this group.
+
+\begin{center}
+\begin{tikzpicture}[scale=2]
+	\draw (0,0) -- (1,0) -- (1,1) -- (0,1) -- (0,0);
+
+	\draw[gray] (1.25,1.25) -- (-0.25,-0.25) node[below left]{$d$};
+	\draw[gray] (1.25,-0.25) -- (-0.25,1.25) node[above left]{$f$};
+	\draw[gray] (0.5,-0.25) -- (0.5,1.25) node[above]{$g$};
+	\draw[gray] (-0.25, 0.5) -- (1.25,0.5) node[right]{$h$};
+
+\end{tikzpicture}
+\end{center}
+\vfill
+
+\problem{}
+Create a multiplication table for all symmetries of a square.
+\vfill
+\pagebreak
+
+\problem{}
+Create a multiplication table for all symmetries of a rhombus.
+\vfill
+\pagebreak
+
+\problem{}
+Find the order of each element in...
+\begin{itemize}
+	\item The group of symmetries of a square
+	\item The group of symmetries of a rhombus
+\end{itemize}
+
 
 \vfill
 \pagebreak
 
-\definition{}
-Recall your tables from \ref{modtables}: \\
-
-\begin{center}
-\begin{tabular}{c | c c c c}
-	+ & 0 & 1 & 2 & 3 \\
-	\hline
-	0 & 0 & 1 & 2 & 3 \\
-	1 & 1 & 2 & 3 & 0 \\
-	2 & 2 & 3 & 0 & 1 \\
-	3 & 3 & 0 & 1 & 2 \\
-\end{tabular}
-\hspace{1cm}
-\begin{tabular}{c | c c c c}
-	\times & 1 & 2 & 3 & 4 \\
-	\hline
-	1 & 1 & 2 & 4 & 3 \\
-	2 & 2 & 4 & 3 & 1 \\
-	3 & 4 & 3 & 1 & 2 \\
-	4 & 3 & 1 & 2 & 4 \\
-\end{tabular}
-\end{center}
-
-Look at these tables and convince yourself that $(\mathbb{Z}/4, +)$ and $( (\mathbb{Z}/5)^\times, \times )$ are the same group. \\
-We say that two such groups are \textit{isomorphic}.
-
-\vspace{2mm}
-
-Intuitively, this means that these two groups have the same algebraic structure. We can translate statements about addition in $\mathbb{Z}/4$ into statements about multiplication in $(\mathbb{Z}/5)^\times$ \\
-
-\pagebreak
-

Advanced/Group Theory/parts/02 isomorphism.tex

@@ -0,0 +1,56 @@
+\section{Isomorphism}
+
+\definition{}
+We say two groups are \textit{isomorphic} if we can create a bijective mapping between them.
+
+\problem{}
+Recall your tables from \ref{modtables}: \\
+\begin{center}
+\begin{tabular}{c | c c c c}
+	+ & 0 & 1 & 2 & 3 \\
+	\hline
+	0 & 0 & 1 & 2 & 3 \\
+	1 & 1 & 2 & 3 & 0 \\
+	2 & 2 & 3 & 0 & 1 \\
+	3 & 3 & 0 & 1 & 2 \\
+\end{tabular}
+\hspace{1cm}
+\begin{tabular}{c | c c c c}
+	\times & 1 & 2 & 3 & 4 \\
+	\hline
+	1 & 1 & 2 & 4 & 3 \\
+	2 & 2 & 4 & 3 & 1 \\
+	3 & 4 & 3 & 1 & 2 \\
+	4 & 3 & 1 & 2 & 4 \\
+\end{tabular}
+\end{center}
+Are $(\mathbb{Z}/4, +)$ and $( (\mathbb{Z}/5)^\times, \times)$ isomorphic? If they are, find a bijection that maps one to the other.
+\vfill
+
+\problem{}
+Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. Show that $f(e_A) = e_B$, where $e_A$ and $e_B$ are the identities of $A$ and $B$.
+\vfill
+
+\problem{}
+Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$ be a bijection. \\
+Show that $f(a^{-1}) = f(a)^{-1}$ for all $a \in A$.
+\vfill
+
+\problem{}
+Let groups $A$ and $B$ be isomorphic, and let $f: A \to B$. Show that $f(a)$ and $a$ have the same order.
+
+\vfill
+\pagebreak
+
+\problem{}
+Find all distinct groups of two elements. \\
+Find all distinct groups of three elements. \\
+Groups that are isomorphic are not distinct.
+
+\vfill
+
+\problem{}
+Show that the groups $(\mathbb{R}, +)$ and $(\mathbb{Z}^+, \times)$ are isomorphic.
+\vfill
+
+\pagebreak

Advanced/Group Theory/parts/03 bonus.tex

@@ -0,0 +1,47 @@
+\section{Bonus}
+
+\problem{}
+Find the inverse of 19 in $\mathbb{Z}/23$ \\
+\hint{Recall the Euclidian Algorithm}
+
+
+\begin{solution}
+	17
+\end{solution}
+\vfill
+
+\problem{}
+Prove Lagrange's theorem:
+
+$$
+	a^p = a \text{ (mod p)}
+$$
+
+\vfill
+
+\problem{}
+Show that $a$ has an inverse mod $m$ iff $\gcd(a, m) = 1$ \\
+
+\begin{solution}
+	Assume $a^\star$ is the inverse of $a \pmod{m}$. \\
+	Then $a^\star \times a \equiv 1 \pmod{m}$ \\
+
+	Therefore, $aa^\star - 1 = km$, and $aa^\star - km = 1$ \\
+	We know that $\gcd(a, m)$ divides $a$ and $m$, therefore $\gcd(a, m)$ must divide $1$. \\
+	$\gcd(a, m) = 1$ \\
+
+	Now, assume $\gcd(a, m) = 1$. \\
+	By the Extended Euclidean Algorithm, we can find $(u, v)$ that satisfy $au+mv=1$ \\
+	So, $au-1 = mv$. \\
+	$m$ divides $au-1$, so $au \equiv 1 \pmod{m}$ \\
+	$u$ is $a^\star$.
+\end{solution}
+
+\vfill
+
+
+\problem{}
+Show that for any integers $a, b, c$, \\
+$\gcd(ac + b, a) = \gcd(a, b)$\\
+
+\vfill