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{\large\textbf{Dartmouth College}\\}
Mathematics 101\\\medskip Homework 4 \small{(due Thursday, Oct 22)}
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\begin{enumerate}

\item Let $G$ be a finite group and $H$ a proper subgroup.  Show that
  $G$ is not the union of conjugates of $H$.  Note the contrast to an
  earlier homework with $G = GL_2(\C)$ and $H$ the group of upper
  triangular matrices.

\item For $n \ge 3$, characterize the center of the symmetric group,
  $S_n$.

\item Show that for $n \ge 5$, the only normal subgroups of $S_n$ are
  $\{e\}$, $A_n$ and $S_n$.  Use this fact to show that for $n\ge 5$,
  $S_n$ is not a solvable group.

\item Let $G$ be a nonabelian group of order $p^3$, $p$ a prime.  Show
  that the commutator of $G$ equals its center.

\item Let $p < q$ be primes, and $G$ a nonabelian group of order $pq$.
  Show that there is an embedding of $G$ into the symmetric group
  $S_q$.

\item Let $p$, $q$ be primes (not necessarily distinct).  Show that
  any group of order $p^2q$ is solvable.



\end{enumerate}


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