Added group theory parts

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
-