Merge branch 'master' of ssh://git.betalupi.com:33/Mark/ormc-handouts

Advanced/Group Theory/parts/01 groups.tex

@@ -6,7 +6,7 @@ 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: $a \ast b = b \ast a$
+	\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 $a$. \\
 	This element is written as $-a$ if our operator is addition and $a^{-1}$ otherwise.
@@ -42,13 +42,59 @@ Make it one by modifying $\mathbb{R}$. \\
 \vfill
 
 \problem{}
-Can you construct a group that contains a single element?
+What is the smallest group we can create?
 
 \begin{solution}
 	Let $(G, \circledcirc)$ be our group, where $G = \{\star\}$ and $\circledcirc$ is defined by the identity $\star \circledcirc \star = \star$
 
 	Verifying that the trivial group is a group is trivial.
 \end{solution}
+\vfill
+
+\problem{}
+Let $G$ be the set of all bijections $A \to A$. \\
+Let $\circ$ be the usual composition operator. \\
+Is $(G, \circ)$ a group?
+
+\vfill
+\pagebreak
+
+\problem{}
+Show that a group has exactly one identity element.
+\vfill
+
+\problem{}
+Show that each element in a group has exactly one inverse.
+\vfill
+
+\problem{}
+Let $(G, \ast)$ be a group and $a, b, c \in G$. Show that...
+\begin{itemize}
+	\item $a \ast b$ and $a \ast c \implies b = c$
+	\item $b \ast a$ and $c \ast a \implies b = c$
+\end{itemize}
+What does this mean intuitively?
+\vfill
+
+\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 $g^n = \overline{0}$ \\
+\hint{$g^n = g \ast g \ast ... \ast g$ $n$ times.}
+
+\vspace{2mm}
+
+The smallest such $n$ defines the \textit{order} of $(G, \ast)$.
+
+\vfill
+
+\problem{}
+What is the order of 5 in $(\mathbb{Z}/25, +)$? \\
+What is the order of 2 in $((\mathbb{Z}/17)^\times, \times)$? \\
+
+\vfill
+
+\problem{}
+Show that if $G$ has four elements, $(G, \ast)$ is abelian.
 
 \vfill
 \pagebreak
@@ -76,20 +122,12 @@ Recall your tables from \ref{modtables}: \\
 \end{tabular}
 \end{center}
 
-Convince yourself that $(\mathbb{Z}/4, +)$ and $( (\mathbb{Z}/5)^\times, \times )$ are the same group. \\
+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 $(\mathbb{Z}/4, +)$ and $( (\mathbb{Z}/5)^\times, \times )$ have the same algebraic structure. We can translate statements about addition in $\mathbb{Z}/4$ into statements about multiplication in $(\mathbb{Z}/5)^\times$ \\
+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 $(\mathbb{Z}/5)^\times$ \\
 
-\problem{}
-Show that a group has exactly one identity element.
-\vfill
-
-\problem{}
-Show that each element in a group has exactly one inverse.
-
-\vfill
 \pagebreak