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\title{Group Theory: Sheet 9}
\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 Suppose that $\alpha: G \rightarrow H$ is a group epimorphism.
Suppose that $K \leq G$ and let ${\overline K}$ denote the
group $\{ (k)\alpha \mid k \in K\}$. Suppose that $L \leq H$.
Let $\widehat L$ denote the group $\{ g \in G \mid
(g)\alpha \in L\}.$ Let $N = \mbox{Ker }\alpha$.
\begin{enumerate}
\item Suppose that $A \leq G$. Show that $\widehat {\overline A} = AN$.
\item Suppose that $B \leq H$. Show that $\overline {\widehat B} = B$.
\item Suppose that $N \leq C \leq D \leq G$. Show that
$|D:C| = |{\overline D}: {\overline C}|$ and that $C \unlhd D$ if and only if
${\overline C} \unlhd {\overline D}$.
\item Suppose that $E \leq F \leq H$. Show that $|F:E|
=|\widehat F: \widehat E|$.
\item Show that ``bar'' and ``hat'' set up an inclusion preserving,
index preserving bijection between the subgroups of $G$ which contain $N$,
and the subgroups of $H$.
\end{enumerate}
\item Suppose that $G$ is a finite group and that $N \unlhd G$.
Let $p$ be a prime number and suppose that $P \in \mbox{Syl}_p(N).$
Show that $G = N_G(P)N$.
\item Suppose that $G$ is a finite group and that $p$ is
a prime number. Let $P$ be a Sylow $p$ subgroup of $G$.
\begin{enumerate}
\item Suppose that $N_G(P) \leq H \leq G$. Prove that $N_G(H) = H$.
\item Suppose that $N_G(P) \leq K \leq G$ and that
$N_G(P) \leq L \leq G$. Furthermore suppose that there is $g \in G$
such that $H^g = K$. Prove that $H = K$.
\end{enumerate}
\end{enumerate}
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