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\prof{Chris Bendel and Peter Cholak} \class{Math 222 - Quiz 9}
\quizdate{Wednesday, March 31} \pagestyle{empty}

\begin{document}


\heading

Be sure to carefully write up your answers.  It is suggested that you
first write out a draft of your proposed questions and then carefully
rewrite that draft to get your final version.  You do \emph{not} have
to write the answers on this sheet of paper.\\


\begin{problem}
  Let $\sigma = \left(
  \begin{array}{ccccccccc}
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
    2 & 3 & 4 & 1 & 6 & 7 & 5 & 9 & 8
  \end{array} \right)$ and  $\tau = \left(
  \begin{array}{ccccccccc}
    1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
    9 & 3 & 4 & 2 & 7 & 5 & 1 & 6 & 8
  \end{array} \right)$.  

\begin{subproblem}
  Express both $\sigma$ and $\tau$ in disjoint cycle notation.
\end{subproblem}

\begin{subproblem}
  Express $\sigma^{-1}, \sigma\tau$ and $\tau\sigma$ in disjoint cycle
  notation.
\end{subproblem}
\begin{subproblem}
  Find the orders of $\sigma$, $\tau$, and $\sigma\tau$.
\end{subproblem}

\begin{subproblem}
  Write $\sigma$ and $\tau$ as a product of transpositions.
\end{subproblem}
\begin{subproblem}
  Find the parity of $\sigma$, $\tau$, $\sigma\tau$ and
  $\sigma\tau^2$.
\end{subproblem}
\end{problem}


\end{document}    
