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\begin{center}
  \textbf{Math 71}\\ Homework Assignment 25 - 29, October 1999
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\begin{enumerate}
\item p. 124: 8, 10, 14
  
\item Let $G$ be a group of order 105.  Show that $G$ has both a
  normal Sylow 5- subgroup and a normal Sylow 7-subgroup.
  
\item Let $G$ be a group of order 48.  Show that $G$ has a normal
  subgroup of order 8 or 16.
  
\item Let $G$ be a group of order 231, and suppose that $G$ has only
  one Sylow 3-subgroup.  Show that $G$ is cyclic.
  
  \bigskip \textbf{A few hints...}
  
  For problem 2:
\begin{enumerate}
  
\item First show that if $H$ is a group of order 35, all its Sylow
  $p$-subgroup s are normal in $H$ (i.e. $n_5 = n_7 = 1$).
  
\item Next show that if $G$ is a group of order 105, for at least one
  of $p=5$ or $p=7$, we have $n_p = 1$.
  
\item For each $p = 5, 7$, let $H_p$ denote a fixed Sylow $p$-subgroup
  of $G$.  Show that $H = H_5H_7$ is a normal subgroup of $G$.
  
\item Let $P$ be any Sylow $p$-subgroup of $G$, $p = 5$ or 7.  Show
  that $P = H_p$.
\end{enumerate}

For problem 3:
\begin{enumerate}
  
\item If there is more than one Sylow 2-subgroup, let $H$ and $K$ be
  any two of them.  Show that $|H\cap K| = 8$.
  
\item Show that $H,K \subset N_G(H\cap K)$.
  
\item Show that $G = N_G(H\cap K)$.

\end{enumerate}



\end{enumerate} 
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