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\parbox{2.25in}{\bf Math 4720 \& 5210}
\hfill {\Large \bf Homework \#2} \hfill
\parbox{2.25in}{\bf \hfill Due: Fri., Sept. $3^\mathrm{rd}$, 2021}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Establishing Order Redux] Recall that $|g|$ denotes the order of $g$ when $g$ is an element of a group. Let $G$ be a group and $a,b \in G$. Prove that $|a| = |bab^{-1}|$ (conjugates have the same order). 

{}{\bf [Grad. Students:]} Also prove that $|ab|=|ba|$.

{\it Note/Hint:} Make sure your proof works for elements of infinite order. Don't assume anything commutes. You may assume this (nearly self-evident) {\bf Lemma:} Let $x,y \in G$ (a group) and suppose that for any $m \in \mathbb{Z}$, $x^m=1$ iff $y^m=1$. Then it follows that $|x|=|y|$. 

\item[\#2 Modding out Happiness] Here we practice some modular arithmetic. Recall that $\mathbb{Z}_n^\times = \{ k \in \mathbb{Z}_n \;|\; \mathrm{gcd}(k,n)=1 \}$ is a group under multiplication mod $n$ whereas $\mathbb{Z}_n$ is a (cyclic) group under addition mod $n$.

\begin{enumerate}[(a)]
\item How many elements of order $8$ are there in $\mathbb{Z}_{23432}$? Find them.

\item How many elements of order $8$ are there in $\mathbb{Z}_{12345}$?

\item Use the (extended) Euclidean algorithm to show that $56 \in \mathbb{Z}_{12345}^\times$ and actually find $56^{-1}$.
\end{enumerate}

For examples of running the (extended) Euclidean algorithm check out answer keys to old Test \#1's found with my {\color{blue}\href{https://billcookmath.com/exams/math3110}{Math 3110 Exams}}. In particular, {\color{blue}\href{https://billcookmath.com/exams/math3110/math3110-spring2021-test1-answer_key.pdf}{Spring 2021 Test \#1 Answer Key Problem 3c}} gives an example.

You can check your work with {\color{blue}\href{https://billcookmath.com/sage/algebra/Euclidean_algorithm.html}{https://billcookmath.com/sage/algebra/Euclidean\_algorithm.html}}

\item[\#3 Dihedral Fun] List all of the cyclic subgroups of $D_{10}$. 

Make a table listing off the possible orders of elements in $D_{10}$ along with how many elements have that order. 

Find $Z(D_{10})$. \qquad {\it Note:} Recall that $Z(G) = \{ g \in G \;|\; gx=xg \mbox{ for all } x\in G\}$ is the {\it center} of $G$. 

{}{\bf [Grad. Students:]} Find $Z(D_{n})$ for all $n \geq 3$. {\it Hint:} You will want to consider $n=2k$ and $n=2k+1$ (even vs. odd) separately.

\item[\#4 A Permuted Problem] For each of the follow permutations $\tau$, determine if there exists some $n$-cycle $\sigma$ such that $\sigma^k = \tau$ for some $k$.

{\it Example:} If $\tau=(13)(24)$, then the answer is ``yes''. Because given $\sigma=(1234)$ we have $\sigma^2=(1234)^2 = (13)(24) = \tau$.

\begin{enumerate}[(a)]
\item $\tau = (12)(34)(56)(78)(9,10)$
\item $\tau = (12)(345)$
\end{enumerate}

\item[\#5 Conjugatin' and Permutatin'] Let $G$ be a group and $g \in G$. Recall that the set of all conjugates of $g$, $\{ xgx^{-1} \;|\; x \in G\}$, is called the {\it conjugacy class} of $g$ (in $G$). 

\begin{enumerate}[(a)]
\item List all of the conjugacy classes of $S_4$. 

{}{\bf [Grad. Students:]} What are the conjugacy classes of $A_4$?

\item When $\sigma \in S_7$, what are the possible orders of $\sigma$? Give an example $\sigma$ for each possible order.

\item Find the smallest positive integer $n$ such that $S_n$ has an element of order 15. What's the smallest $n$ can be if we wish to have an element of order 16?
\end{enumerate}



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