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\noindent
\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{1.95in}{\bf \hfill Due: Fri., Mar. 1${}^\mathbf{st}$, 2024}

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%\noindent {\bf\large Name:} \quad \underline{\large\qquad This is my Name \qquad}

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\noindent
Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force

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\begin{description}
\item[\#1]  Let $\sigma=(12)(456)$. Consider the permutations $\tau$ in the table below.\\ \force \hfill $\Longrightarrow$ Remember to give {\it simplified good manners} answers!

Fill out the following table:

\begin{tabular}{|c|c|c|c|} \hline
$\tau=$ & $(12)(345)(246)(13)$ & $(146)(35)(2534)$ & $(245)(1234)$ {\Large\strut} \\ \hline
$\tau$ simplified (as disjoint cycles): & & & {\Large\strut} \\ \hline
The order of $\tau$: $|\tau|=$ &  & & {\Large\strut} \\ \hline
The inverse of $\tau$: $\tau^{-1}=$ & & & {\Large\strut} \\ \hline 
$\tau$ as a product of transpositions: & & & {\Large\strut} \\ \hline
$\tau$ conjugated by $\sigma$: $\sigma \tau \sigma^{-1}=$ & & & {\Large\strut} \\ \hline
A power of $\tau$: $\tau^{33}=$  & & & {\Large\strut} \\ \hline
\end{tabular}
  
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\item[\#2] Orders in $S_n$.
   \begin{enumerate}[(a)]
   \item What are the orders of the elements in $S_8$? Give an example of an element with each order.
   
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   {\it Note:} Recall that orders of permutations depend on their cycle structure and that cycle structures correspond to partitions. There are 22 partitions of 8. Typing ``partitions of 8'' into Wolfram Alpha (and asking it to show ``More'') you can get a complete list. That said, many partitions yield redundant element orders. You don't need to catalog of all possible structures. Just give an example for each {\it distinct} element order. [For example: $(12)$ has order 2, so examples like $(12)(34)$ and $(12)(34)(56)$ -- also order 2 -- are redundant.]
   
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   \item A 60 cycle in $S_{60}$ has order 60. But elements of order 60 show up much sooner than that. What is the smallest $n$ such that $S_n$ has an element of order $60=2^2\cdot 3\cdot 5$? Give an example of such an element.

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   \item When does order 9 first show up in a  permutation group $S_n$? Give an example of such an element and explain why 9 does not show up sooner.

   \end{enumerate}
 
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\item[\#3 RESUBMIT] Type up Homework \#4 Problem \#4 part (a) and its solution in \LaTeX. 

\fbox{\parbox{6.5in}{
Show the relation: $x \sim y$ iff $y$ is a conjugate of $x$ (i.e., there exists $g \in G$ such that $gxg^{-1}=y$) is an {\it equivalence relation}.
   
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In other words, show\dots
\begin{itemize}
\item $\sim$ is reflexive: for all $x \in G$, we have $x \sim x$; 
\item $\sim$ is symmetric: if $x \sim y$, then $y \sim x$; \quad and 
\item $\sim$ is transitive: if $x \sim y$ and $y \sim z$, then $x \sim z$.
\end{itemize}   
}}

     When typing this problem up, please write carefully: Restate the problem. Write in complete sentences.  

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