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\noindent
\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{1.95in}{\bf \hfill Due: \sout{Fri., Oct. 4${}^\mathbf{th}$, 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 $\sigma=(152)(46)$. 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
$\tau=$ & $(1536)(24)(645)$ & $(23)(175)(1542)$ & $(34)(156)(246)(35)$ {\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 $\tau$: $\tau^{23}=$  & & & {\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 $S_9$? 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 30 partitions of 9. Typing ``partitions of 9'' 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 cycle structures. Just give an example for each {\it distinct} element {\it 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 90 cycle in $S_{90}$ has order 90. But elements of order 90 show up much sooner than that. What is the smallest $n$ such that $S_n$ has an element of order $90=2\cdot 3^2\cdot 5$? Give an example of such an element.

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   \item When does order 8 first show up in a  permutation group $S_n$? Give an example of such an element and explain why 8 does not show up sooner.

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

\fbox{\parbox{6.5in}{
 Let $x,y \in G$ (for some group $G$). If there exists some $g \in G$ such $gxg^{-1}=y$, we say $x$ and $y$ are {\it conjugates}.

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Problem: Use induction to show $(gxg^{-1})^k=gx^kg^{-1}$ for any $k \in \mathbb{Z}_{\geq 0}$. 

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   {\it Note:} Consequently, $(gxg^{-1})^n=e$ iff $gx^ng^{-1}=e$ iff $x^n=g^{-1}eg=e$, so conjugates have the same order.
}}

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

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