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  \textbf{Math 71}\\ Homework 6
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\begin{enumerate}

\item page 122: 10, 14

\item page 165: \#2,3(b,e)

\item Let $G$ be a group of order 48.  Show that $G$ has a normal
    subgroup of order 8 or 16.  Some hints are provided.
    \begin{enumerate}
    \item If there is more than one Sylow 2-subgroup, let $H$ and $K$
      be any two (distinct) Sylow 2-subgroups.  Show that $|H\cap K| =
      8$.

      \item Show that for the $H$ and $K$ above, that $H, K \subseteq
        N_G(H\cap K)$.

\item Show that $G = N_G(H\cap K)$.
    \end{enumerate}

\item Let $G$ be a group of order $231 = 3\cdot 7\cdot 11$, and suppose
  that there is only one Sylow 3-subgroup.  Show that $G$ is cyclic.


\end{enumerate}




 

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