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\noindent
\parbox{2.25in}{\bf Math 3110-101}
\hfill {\Large \bf Homework \#5} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Mar. $5^\mathrm{th}$, 2021}

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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 $\sigma=(123)(45)$. Fill out the following table:

\begin{tabular}{|c|c|c|c|} \hline
$\tau=$ & $(1432)(56)(254)$ & $(1234)(1423)(246)$ & $(12)(345)(1357)$ {\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 $\tau$: $\tau^{99}=$  & & & {\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 $S_7$? Give an example of an element with each order.
   
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   {\it Note:} It might be helpful to know that there are 15 partitions of the number 7. Each partition corresponds to a different cycle type. Specifically, $7=6+1=5+2=5+1+1=4+3=4+2+1=4+1+1+1=3+3+1=3+2+2=3+2+1+1= 3+1+1+1+1=2+2+2+1=2+2+1+1+1=2+1+1+1+1+1=1+1+1+1+1+1+1$.

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   \item Does $S_{10}$  have an element of order 30? If so, find one. If not, explain why not.
   
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   \item Does $S_{10}$  have an element of order 25? If so, find one. If not, explain why not. 
   \end{enumerate}
 
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\item[RESUBMIT] Type up Homework \#4 Problems \#4 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}.
   \begin{enumerate}[(a)]
   \item Let $y=gxg^{-1}$ for some $g \in G$. Show that $|x|=|y|$ (i.e. conjugates have the same order).  
   
{\color{red}{\bf Note:}} You will need to know that $(gxg^{-1})^k = gx^kg^{-1}$ for every non-negative integer $k$. I let you get away with informal justifications on original submissions. This time I would like you to include a careful proof of this fact (use induction on $k$).
   
   \item Prove or give a counterexample: $\langle x \rangle = \langle gxg^{-1} \rangle$ (where $x,g \in G$). 
   
            In other words, is it true or not that conjugates generate the same cyclic subgroup?
   \end{enumerate}
}}

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

\end{description}

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