Finished cryptography handout

Advanced/Cryptography/parts/2 groups.tex

@@ -45,55 +45,11 @@ What is the smallest group we can create?
 
 	Verifying that the trivial group is a group is trivial.
 \end{solution}
-\vfill
 
+
+\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{}
-%Show that $(\mathbb{Z}_n^\times, \times)$ is a group for any $n \in \mathbb{Z}^+$.
-%\vfill
-
-%\problem{}
-%Let $(G, \ast)$ be a group and $a, b, c \in G$. Show that...
-%\begin{itemize}
-%	\item $a \ast b = a \ast c \implies b = c$
-%	\item $b \ast a = c \ast a \implies b = c$
-%\end{itemize}
-%This means that we can \say{cancel} operations in groups, much like we do in algebra.
-%\vfill
-%\pagebreak
-
-
-
-
-% \problem{}
-% Let $G$ be the set of all bijections $A \to A$. \par
-% Let $\circ$ be the usual composition operator. \par
-% Is $(G, \circ)$ a group?
-%
-% \vfill
-%
-% \definition{}
-% Note that our definition of a group does \textbf{not} state that $a \ast b = b \ast a$. \par
-% Many interesting groups do not have this property.
-% Those that do are called \textit{abelian} groups. \par
-%
-% \vspace{2mm}
-%
-% One example of a non-abelian group is the set of invertible 2x2 matrices under matrix multiplication. In this handout, all % groups are abelian.
-%
-%
-%
-% \problem{}
-% Show that if $G$ has four elements, $(G, \ast)$ is abelian.
 
 \problem{}
 Let $(G, \ast)$ be a group with finitely many elements, and let $a \in G$. \par