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\title{Group Theory: Math30038, Sheet 8}
\date{GCS}
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\begin{enumerate}
\item Let $P$ be a Sylow $p$-subgroup of the finite group $G$.
Suppose that $N \unlhd G$. 
\begin{enumerate}
\item Show that $P \cap N \in \mbox{Syl}_p(N).$ 
\item Show that $PN/N \in \mbox{Syl}_p(G/N).$
\end{enumerate}
\item Show that every group of order $15$ must be abelian.
\item Show that every group of order $35$ must be abelian.
\item Show that there is no non-abelian finite simple group
of order less than 60.
\item Let $p$ and $q$ be distinct prime numbers.
\begin{enumerate}
\item Show that a group of order $pq$ can not be simple.
\item Show that a group of order $p^2q$ can not be simple.
\item Show that a group of order $p^2q^2$ can not be simple.
\end{enumerate} 
\end{enumerate}  
\end{document}
