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\begin{flushright}
  Your Name Here
\end{flushright}




\begin{center}
  \textbf{Math 71}\\ Homework 6
\end{center}



\begin{enumerate}
\item Show that every nonabelian group of order 6 has a nonnormal
  subgroup of order 2.  Use this to classify all groups of order 6.
  \textbf{Note:} This is the exercise you prove to justify the
  statement we have been using all term, that there are exactly two
  isomorphism classes of groups of order 6.  So do not invoke that
  result here.

\item Show that for $n \ge 3$ and $p$ an odd prime, that every Sylow
  $p$-subgroup of $D_{2n}$ is cyclic and normal in $D_{2n}$.

\item List all the Sylow 2-subgroups of $S_4$, and given two of them,
  find an element $\sigma$ is $S_4$ which conjugates one to the other.
  \textbf{Hint}: Can you find a permutation $\tau$ so that
  $\tau (a_1\,a_2\, \dots \, a_r) \tau^{-1}= (b_1\,b_2\,\dots\,b_r)$?
  Parts of \S4.3 that we did not cover in class have some interesting
  results!
  
\item Let $G$ be a group of order $105 = 3\cdot 5\cdot 7$.  For each
  prime $p$ dividing 105, let $H_p$ denote a Sylow $p$-subgroup of
  $G$.
  \begin{itemize}
  \item Show that $n_5 = 1$ or $n_7 = 1$.
  \item Use that to conclude that $H_5H_7$ is a normal subgroup of $G$
    and that $n_5 = n_7 =1$.
  \item Show that if $n_3=1$, then $G$ is cyclic.
  \end{itemize}

\item Determine all the distinct isomorphism classes of abelian groups
  of order $9801= 3^4\cdot 11^2$.  Give two lists, one using
  elementary divisors and the other using invariant factors.
\end{enumerate}

 

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