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\begin{center}
{\large\textbf{Dartmouth College}\\}
Mathematics 101\\\medskip Homework 4 \small{(due Wednesday, Oct 24)}
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\begin{enumerate}



\item Show that the following three statements are equivalent. The
  second is the Feit-Thompson theorem.
  \begin{enumerate}
  \item A finite non-abelian simple group has even order.
    \item A simple group of odd order is isomorphic to $\Z/p\Z$ where
      $p$ is a prime.
      \item Every group of odd order is solvable.
  \end{enumerate}



\item  Let $G$ be a finite group and $H$ a proper subgroup.  Show that
  $G$ is not the union of conjugates of $H$.

\item Let $G$ be a finite simple group containing an element of order
  21.  Show that every proper subgroup $H$ of $G$ satisfies
 $[G : H ] \ge 7$ and show $|G| \ge 168$.

\item  Let $p$ be a prime, and $G$ a $p$-group, i.e., $|G| = p^m$ for
  some $m \ge 1$.  

  \begin{enumerate}
  \item Suppose that $H$ is a nontrivial normal subgroup of
  $G$.  Show that $H \cap Z_G \ne \{e\}$ where $Z_G$ is the center of
  $G$.  In particular, every normal subgroup $H$ of order $p$ is
  contained in the center.
  \item  If $G$ is nonabelian of order $p^3$, show that $Z_G = G'$
    (the commutator), and they are both cyclic of order $p$.
  \end{enumerate}


\end{enumerate}


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