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\noindent{\Large{\textsc{Modern Algebra \hfill Assignment 10.1  \\ Spring 2019 \hfill Due April 10}}}

\large

\vspace{1cm}


\begin{exercise}
If $\sigma \in S_n$ and $(i_1 \, i_2 \, \cdots \, i_r) \in S_n$ is an $r$-cycle, show that
\[
\sigma (i_1 \, i_2 \, \cdots \, i_r) \sigma^{-1} =  (\sigma(i_1) \, \sigma(i_2) \, \cdots \, \sigma(i_r)).
\]
Use this and the previous exercise to prove that all $r$-cycles are conjugate in $S_n$.  [{\em Suggestion:}  Prove the two permutations are equal by evaluating them both at every point, noting that every $j \in I_n$ can be written $j=\sigma(i)$ for some $i \in I_n$.]
\end{exercise}

Given $\sigma \in S_n$, write $\sigma = c_1 c_2 \dots c_k$, where the $c_j$ are disjoint cycles and every $i \in I_n$ occurs in exactly one $c_j$ (we include 1-cycles).  Reorder the cycles in $\sigma$ so that $\ell(c_1) \ge \ell(c_2) \ge \cdots \ge \ell(c_k)$, where $\ell(c)$ denotes the length of a cycle $c$.  Finally, let
\[
\lambda(\sigma) = (\ell(c_1) , \ell(c_2), \cdots , \ell(c_k)).
\]

\begin{exercise}  
Prove that $\lambda(\sigma)$ is a {\em partition} of $n$, i.e.\ that the sum of the entries of $\lambda(\sigma)$ is $n$.
\end{exercise}

\begin{exercise}
Prove that $\lambda(\tau \sigma \tau^{-1}) = \lambda(\sigma)$ for all $\tau \in S_n$.  [{\em Suggestion:}  Write $\sigma$ as a product of disjoint cycles and apply Exercise 1.]
\end{exercise}

\begin{exercise}
Prove that the converse of the result in the previous exercise also holds.  That is, show that if $\tau \in S_n$ and $\lambda(\sigma) = \lambda(\tau)$, then $\sigma$ and $\tau$ are conjugate in $S_n$.  [{\em Suggestion:}  If you write down the cycle decompositions of $\sigma$ and $\tau$, and appeal to Exercise 1, you can easily construct a permutation that conjugates $\sigma$ into $\tau$.]
\end{exercise}

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