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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 10.1  \\ Spring 2019 \hfill Due March 29}}}

\large

\vspace{1cm}

\begin{exercise}
Let $\gamma =(i_1 \, i_2 \, \cdots \, i_r) \in S_n$ be an $r$-cycle.
\begin{itemize}
\item[\bf a.]  Prove that $\epsilon(\gamma) = (-1)^{r+1}$.

\item[\bf b.]  If $\sigma \in S_n$, show that
\[
\sigma (i_1 \, i_2 \, \cdots \, i_r) \sigma^{-1} =  (\sigma(i_1) \, \sigma(i_2) \, \cdots \, \sigma(i_r)).
\]
\end{itemize}
\end{exercise}


\begin{exercise}
Determine the parity of each of the following permutations.
\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 $G$ be a finite group of order $n$.  Given $a \in G$, let $\lambda_a \in \operatorname{Perm}(G) \cong S_n$ denote left translation by $a$.  If $a$ has order $m$, prove that $\epsilon(\lambda_a)= (-1)^{n(m+1)/m}$.  [{\em Suggestion:}  Show that the cycles of $\lambda_a$ are the right cosets of $\langle a \rangle$ in $G$.]
\end{exercise}



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