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\begin{document}

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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 9.1  \\ Spring 2023 \hfill Due March 22}}}

\large

\vspace{1cm}

\begin{exercise}
Express each of the following permutations as products of disjoint cycles (you may feel free to omit 1-cycles) and as products of transpositions.

\begin{itemize}

\item[\bf a.]  $\displaystyle \left( \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\  1 & 5 & 4 & 2 & 8 & 7 & 6 & 3 \end{matrix} \right)$

\item[\bf b.]  $\displaystyle \left( \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\  10 & 9 & 1 & 5 & 2 & 7 & 8 & 6 & 4 & 3 \end{matrix} \right)$

\item[\bf c.]  $\displaystyle \left( \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\  7 & 6 & 8 & 1 & 9  & 5  & 4 & 2 & 3 \end{matrix} \right)$

\item[\bf d.]  $\displaystyle \left( \begin{array}{ccccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 \\  1 & 2 & 11 & 9 & 10 & 8  & 6 & 7 & 4  & 5 & 3  \end{array} \right)$

\end{itemize}

\end{exercise}

\begin{exercise}
Let $\sigma \in S_n$ and suppose $\sigma = c_1 c_2 \cdots c_k$, where $c_1, c_2, \ldots, c_k$ are disjoint cycles.

\begin{itemize}

\item[\bf a.]  Explain why $\sigma^m = \textrm{Id}$ if and only if $c_i^m = \textrm{Id}$ for all $i$.

\item[\bf b.]  If $\ell(c)$ denotes the length of a cycle $c$, prove that
\[
|\sigma| = \operatorname{lcm}(\ell(c_1), \ell(c_2), \ldots ,\ell(c_k)).
\]

\end{itemize}

\end{exercise}

\begin{exercise}
Determine the order of each permutation in Exercise 1.
\end{exercise}


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