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\begin{center}
  {\large\textbf{Dartmouth College}\\} Mathematics 101\\\medskip
  Homework 5 \small{(due Wednesday, October 31)}
\end{center}

\begin{enumerate}
\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 to give an alternate proof to
  Lang's that $S_n$ is not solvable for $n\ge 5$.  This fact is key in
  showing that the general polynomial of degree $n \ge 5$ is not
  solvable by radicals.



\item For $p$ and $q$ distinct primes, show that any group of order
  $p^2 q$ is solvable.



  


\item Semidirect products.  We shall show in class that $Aut(\Z_n)
  \cong \Z_n^\times$.

  \begin{enumerate}
  \item Suppose that $H_1$, $H_2$ and $K$ are groups, $\sigma: H_1 \to
    H_2$ is an isomorphism, and \newline $\psi:H_2 \to Aut(K)$ a
    homomorphism, so that $\varphi = \psi\circ \sigma: H_1 \to Aut(K)$
    is also a homomorphism.  Show that $K \rtimes_\varphi H_1 \cong K
    \rtimes_\psi H_2$.
  
  \item Suppose that $H$ and $K$ are groups and $\varphi, \psi:H \to
    Aut(K)$ are monomorphisms with the same image in $Aut(K)$.  Show
    that there exists a $\sigma \in Aut(H)$ such that $\psi =
    \varphi\circ \sigma$.

  \item Suppose that $H$ and $K$ are groups, $\varphi, \psi:H \to
    Aut(K)$ are monomorphisms, and $Aut(K)$ is finite and cyclic.
    Show that $\varphi$ and $\psi$ have the same image in $Aut(K)$.

  \item Let $p < q$ be primes with $p \mid (q-1)$. Let $H$ and $K$ be
    cyclic groups of order $p$ and $q$ respectively.  Let $\varphi,
    \psi: H \to Aut(K)$ be nontrivial homomorphisms. Observing that
    $Aut(K)$ is cyclic, show that $K\rtimes_\varphi H \cong
    K\rtimes_\psi H$.
  \end{enumerate}

\item Let $p < q$ be primes, and let $G$ be a group of order $pq$. We
  know from class that $G \cong \Z_q\rtimes_\varphi \Z_p$ for some
  $\varphi: \Z_p \to Aut(\Z_q)$.  By analyzing all possible $\varphi$,
  find (up to isomorphism) all groups of order $pq$.  If $p=2$,
  describe them without using semidirect products.


\end{enumerate}

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