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\title{Group Theory: Extra Exercises}
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\begin{document}
\begin{enumerate}
\item Suppose that $G$ is a group.
\begin{enumerate} 
\item[(a)] Define a relation $\sim$ on $G$ by $x \sim y$ if and only
if there exists $g \in G$ such that $g^{-1} x g = y.$ Prove that this
is an equivalence relation on $G$ (see section 1.18 in the book). 
The equivalence classes of this equivalence relation are called 
the conjugacy classes of $G.$
\item[(b)] Prove that $G$ is an abelian group if and only if each of
its conjugacy classes has size 1.
\item[(c)] Calculate the conjugacy classes of $S_3,$ the symmetric group
on three letters (see section 5.3 in the book).
\item[(d)] Give an example of a group which has infinitely many
distinct conjugacy classes.
\end{enumerate}
\item Suppose that $G$ is a group. A subgroup $N$ of $G$ is said
to be {\em normal} in $G$ if and only if whenever $x \in G,$ then $xN = Nx.$
\begin{enumerate}
\item[(a)] Show that the group $S_3$ has a normal subgroup of
size $3.$
\item[(b)] Prove that the subgroup $M$ of $G$ is normal in $G$
if and only if $x^{-1}Nx = N$ for every $x \in G.$
\item[(c)] Suppose that $N$ is a normal subgroup of $G,$
that $C$ is a conjugacy class of $G$ (see question 1), and that
$N \cap C \not = \emptyset.$ Prove that $C \subseteq N.$
\end{enumerate}
\item Suppose that $G$ is a group, that $x \in G$ and that
$C$ is the conjugacy class of $G$ which contains $x.$ 
Let $C_G(x) = \{ g \mid g \in G,\ g^{-1}xg = x\}$
(this set is called the {\em centralizer of $x$ in $G$}).
\begin{enumerate}
\item[(a)] Show that $C_G(x)$ is a subgroup of $G.$
\item[(b)] Define Let $S = \{ C_G(x)a \mid a \in G\}$ be the set
of right cosets of $C_G(x)$ in $G.$ Attempt to define a map
$\theta : S \rightarrow C$ by $C_G(x)z \mapsto z^{-1}xz.$ 
{\em You need to be careful here. You need to verify that
if $C_G(x)z_1 = C_G(x)z_2,$ then $z_1^{-1}xz_1 = z_2^{-1}xz_2,$ otherwise
this definition makes no sense. Think about this remark carefully, and 
make sure you understand its significance. This is a subtle 
and important point.}
\item[(c)] Show that the map $\theta$ of part (b) is bijective.
\item[(d)] Show that if $G$ is a finite group, then 
$|C| |C_G(x)| = |G|.$
\end{enumerate}
\end{enumerate}
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