\chapter{Appendix - Group Theory Background}

\begin{exercise}{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be
\item Show that the set of all permutations on $n$ elements forms a
group with the group operation of function composition.
\item Show that any permutation can be written as a composition of
disjoint cycles.
\item Show that any $m$-cycle can be written as a composition of
transpositions.
\ee
\end{exercise}

\oneperpage

\begin{exercise}
What is the order of $S_n$?
\end{exercise}

\oneperpage

\begin{exercise} 
%---puts blank space so itemized list doesn't start right after 'Exercise'
{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be
\item Show that an $n$-cycle can be written as the composition of
$n-1$ transpositions. Thus a $3$-cycle is an even permutation and a
$4$-cycle is an odd permutation!
\item Show that the group of even permutations is a subgroup of
$S_n$. \ee
\end{exercise}

\oneperpage

\begin{exercise}
What is the order of $A_n$?
\end{exercise}

\oneperpage

\begin{exercise}
Show that if we let $a$ represent a reflection along a line passing
through the polygon's center and a vertex, and $b$ a rotation of
$2\pi/n$ around its center, then
$$D_n=\{1, b,\ldots, b^{n-1}, ab,\ldots, ab^{n-1}\}$$
\end{exercise}

\oneperpage

\begin{exercise}
Show that  in $D_n$ as above, we have $ab=b^{n-1}a$, and thus $D_n$
is not abelian for $n>2$.
\end{exercise}

\oneperpage

\begin{exercise}
Show that $D_n$ is isomorphic to a proper subgroup of $S_n$.
\end{exercise}

\oneperpage

\begin{exercise}
Under what conditions, if ever, is
$D_n$ is isomorphic to a subgroup of $A_n$?
\end{exercise}

\oneperpage

\begin{exercise}
Let $g, g'\in G$. Then either $gH=g'H$ or $gH\cap g'H=\emptyset$.
\end{exercise}

\oneperpage

\begin{thm}[Lagrange's Theorem]
\index{Lagrange's Theorem} 
Let $G$ be a finite group, and $H$ a
subgroup. Then the cardinality $|H|$ of $H$ divides the cardinality
$|G|$of $G$ and
\[
[G:H]=\frac{|G|}{|H|}
\]
\end{thm}

\oneperpage

\begin{thm}
Let $H\lhd G$ be a  normal subgroup. Then its left and right cosets
coincide for all $g\in G$, in other words $gH=Hg$ for all $g\in G$.
\end{thm}

\oneperpage

\begin{thm}
An onto homomorphism $f:G\to H$ is an isomorphism if and only if
$\kernel f=\{1_G\}$.
\end{thm}

\oneperpage

\begin{thm}
Let $f:G\to H$ be a homomorphism from a group $G$ to a group $H$,
then $\kernel f \lhd G$ and $f(G)<H$.
\end{thm}

\oneperpage

\begin{thm}[First isomorphism theorem]
\index{isomorphism theorem, first} 
Let $f:G\to H$ be a homomorphism
with $\kernel f=N$. Then $f(H)\cong G/N$.
\end{thm}

\oneperpage

\begin{thm}
A cyclic group that is infinite is isomorphic to $\Z$.
\end{thm}

\oneperpage

\begin{thm}
A finite cyclic group of order $n$ is isomorphic to $\Z_n$, the
integers with addition $\mod n$.
\end{thm}

\oneperpage

\begin{exercise}
{\rule{0cm}{0.1cm}}\newline\vspace*{-0.5cm}
\be
\item Verify that the dihedral group
$D_n=\{1, b,\ldots, b^{n-1}, ab,\ldots, ab^{n-1}\}$ is generated by
$\{a,b\}$.
\item Show that the symmetric group $S_n$, for $n\geq 2$, is generated by the set
of $2$-cycles: $\{(12),(23),\ldots,(n-1,n)\}$.
\item Show that the symmetric group $S_n$, for $n\geq 2$, is generated by the
pair of cycles $(12)$ and $(12\ldots n)$.
\ee
\end{exercise}

\oneperpage

\begin{thm}[Classification of Finitely Generated Abelian Groups]
\index{classification of finitely generated abelian groups}\index{abelian group!finitely generated, classification}
Let $G$ be a finitely generated abelian
group. Then $G$ is isomorphic to:
\[
H_0\oplus H_1\oplus\ldots\oplus H_m
\]
where $H_0$ is a free abelian group, and $H_i\cong \Z_{p_i}$
($i=1,\ldots,n$) where $p_i$ is a power of a prime. The rank of
$H_0$ is unique and is called the \textbf{rank}\index{rank} of $G$.  The orders
$p_1,\ldots,p_m$ are also unique up to reordering.
\end{thm}

\oneperpage

\begin{thm}
$G'\lhd G$, and is the smallest subgroup for which $G/G'$ is
abelian. In other words, if there is a subgroup $N\lhd G$ such that
$G/N$ is abelian, then $G'\subset N$.
\end{thm}

\oneperpage

\begin{thm} 
Isomorphic groups have isomorphic
abelianizations.
\end{thm}

\oneperpage

\begin{exercise}
Confirm that the lists of generators and relations given above
completely determine the groups.
\end{exercise}

\oneperpage

\begin{thm} Suppose that $L$ is a set and $R$ is a collection of
words in $L$.  Then there is a group whose presentation is
$\left<L|R\right>$.\end{thm}

\oneperpage

\begin{exercise}
What is a group presentation for an arbitrary finitely generated
abelian group? for the symmetric group?
\end{exercise}

\oneperpage

\begin{thm} 
For $m\neq n$, $F_m\neq F_n$.
\end{thm}

\oneperpage

\begin{lem} 
We have a natural injections $G\to G*H$ and $H\to G*H$
so that $G$ and $H$ may be viewed as subgroups of $G*H$.
\end{lem}

\oneperpage

\begin{thm} Let $G$ and $H$ be disjoint groups.  Each element in
$G*H$ has a unique expression of the form $g_1h_1\cdots g_nh_n$
where $g_1,\ldots,g_n\in G$, $h_1,\ldots,h_n\in H$, and $g_1$ and
$h_n$ are allowed to be the identity, but no other letter
is.\end{thm}

\oneperpage

