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\title{Group Theory: Sheet 8}
\author{\copyright G. C. Smith 2001}
\date{}
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\maketitle


\textit{The copyright is waived for non-profit making educational
purposes.}

{\it The course web site is available via 
{\tt http://www.bath.ac.uk/$\sim$masgcs/}}

\begin{enumerate}
\item You have 10 beads, 6 are black and 4 are white. How many essentially
different necklaces can you make?
\item (repeat) You have 24 beads to thread on to a circular string to make
a necklace. There are 8 blue beads, 6 red beads, 4 white beads,
2 black beads and 4 green beads. How many essentially different
necklaces can you make? (You may need a calculator, or if that is
too dreadful a thought, you can give a formula).
\item (repeat) Let $G$ be a finite group. It turns 
out that the average (over all
group elements)
size of the centralizer of an element is the number of conjugacy classes
of the group. Why?
\item Suppose that $G$ is a group of order 1,000. Show that $G$ must have
a normal subgroup $N$ with $1 \not = N  \not = G$. 
\item Let $H$ be a subgroup of finite index in a group $G$.
\begin{enumerate}
\item[(a)] Let 
$\Omega = \{ H^x \mid x \in G\}$. Let $G$ act on $\Omega$ by conjugation.
Deduce that $\Omega$ is finite.
\item[(b)] Let
\[ K = \bigcap_{g\in G} H^g.\]
Show that $K$ is a normal subgroup of $G$ and that $|G:K| < \infty.$
\end{enumerate}
\item Let $G$ be a finite group, $p$ be a prime number and suppose
that $P \in \mbox{Syl}_p(G).$
\begin{enumerate}
\item[(a)] Show that $P \cap N \in \mbox{Syl}_p(G)$.
\item[(b)] Show that $PN/N \in \mbox{Syl}_p(G/N)$.
\end{enumerate}
\end{enumerate}
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