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\begin{document}
\title{6320-001 - Spring 2021 - Week 3 (2/02, 2/04)}
\maketitle
\section{This week in 6320}
{\bf 2/02} -- After quickly hitting Cayley's theorem (which we didn't get to last week), we'll discuss normal subgroups and quotient groups in a bit more detail, summarize the Jordan-Holder program, and then give a counting argument with conjugacy classes to see that $A_5$ is a simple group.
{\bf 2/04} -- The Sylow theorem(s)
{\bf Key concepts:} {\it Cayley's theorem, normalizer, normal subgroups, simple groups, the alternating group, Sylow theorems }
\section{Comments and suggested reading}
\begin{itemize}
\item {\bf Dummit and Foote} -- 3.4-3.5, 4.2, 4.4-4.5.
\item In addition to the homework below , you are always encouraged (but not required) to work as many supplementary exercises as you have time for from the suggested reading sections!
\end{itemize}
\section{Preview of next week}
\noindent{\it (This is subject to change depending on how far we get this week!)}
\noindent Simplicity of $A_n, n \geq 5$ (4.6) and semi-direct products (5.5).
\section{Homework}
\noindent {\bf Due Tuesday, February 9 at 11:59pm on Gradescope}
\noindent \emph{All solutions must be typeset using TeX and submitted via Gradescope; handwritten or late submissions will not be accepted. All exercises and problems submitted must start with the statement of the exercise or problem.} \\
\noindent You may work in groups, but you must write up your final solutions individually. Any instances of academic misconduct will be taken very seriously.\\
\noindent\emph{Justify your answers carefully!}\\
\subsection{Exercises} {\it Complete and turn in ALL exercises}: \hfill\\
\noindent Grading scale (for each part of an exercise): \\
3 points -- A correct, clearly written solution\\
2 points -- Right idea, but a minor mistake or not clearly argued\\
1 point -- Some progress but multiple minor mistakes or a major mistake\\
0 points -- Nothing written, totally incorrect, or no substantive progress made towards a solution.\\
\noindent{\bf Exercise 1.}
({\bf DF} Exercise 3.3.7) Let $M$ and $N$ be normal subgroups of $G$ such that $G=MN.$ Show that the natural map
\[ G/(M \cap N) \rightarrow G/M \times G/N \]
is an isomorphism.
\hfill
\hfill\\
\noindent{\bf Exercise 2.}
({\bf DF} Exercises 3.5.3-4)
\begin{enumerate}
\item Show that $S_n = \langle (i,i+1) \, | \, 1\leq i \leq n-1 \rangle$.
\item Show that $S_n = \langle (1\, 2), (1\, 2\, 3\, \ldots n) \rangle.$
\end{enumerate}
\hfill\\
\noindent{\bf Exercise 3.}
\begin{enumerate}
\item Let $G$ be a finite group and $p$ the smallest prime dividing $|G|$. Show that if $H\leq G$ is such that $[G:H]=p$ then $H \lhd G$.
\item Give an example of a non-abelian $G$ and a subgroup $H$ where part (1) applies in each of the following cases:
\begin{enumerate}
\item $[G:H]=p=2$
\item $[G:H]=p>2$
\end{enumerate}
\end{enumerate}
\hfill\\
\noindent{\bf Exercise 4.}
Find a formula for the size of each conjugacy class in $S_n$. (Recall that a conjugacy class in $S_n$ is uniquely determined by the number of $k$-cycles $m_k$ for each $1\leq k \leq n$ in its cycle decomposition). (Hint: If you get stuck or want to check your formula, see {\bf DF} Exercise 4.3.33).
\hfill\\
\noindent{\bf Exercise 5.}
\begin{enumerate}
\item Show that if $|G|=p^n$ for $p$ prime then $Z(G)\neq \{e\}.$
\item Deduce Cauchy's theorem (if $p$ prime divides $|G|$ then $G$ has an element of order $p$) from the above and Sylow's theorem.
\end{enumerate}
\subsection{Problems} {\it Attempt as many as you have time for, but only turn in one (of your choice).} \hfill\\
{\bf Grading scale }(for the problem you turn in): \\
10 points - A correct, complete, and clearly written solution. \\
8 points - Right idea, but one or two minor mistakes or not clearly argued.\\
5 points - Some progress but several minor mistakes or a major mistake. \\
0 points - Nothing written, totally incorrect, or no substantive progress made.\\
{\bf Revision policy: }{\it If you score at least 5 points on the problem you turn in then you will be allowed to submit {\bf one} revision to your solution before March 4th (the day of the midterm). If the revision is correct, complete, and clearly written then your mark will change to 9 points. This policy only applies to the problem you submit, not to the exercises in the previous section. }
\hfill
\noindent{\bf Problem 1.}
Let $X$ be a conjugacy class of even permutations in $S_n$. Then, $X$ is a disjoint union of conjugacy classes in $A_n$. How many are there? Your answer should break into two cases based on a simple condition on the cycle decomposition. (\emph{Hint}: This is outlined in {\bf DF} Exercises 4.3.19-21, but try thinking about it on your own first!)
\hfill \\
\noindent{\bf Problem 2.}
Show that for $n \neq 6$ every automorphism of $S_n$ is inner. (\emph{Hint}: You can use the steps outlined in {\bf DF} Exercise 4.4.18, but try thinking about it on your own first!)
\hfill \\
\noindent{\bf Problem 3.}
({\bf DF 4.5.19-23}). Show there is no simple group of order $n$ for
\begin{enumerate}
\item $n=6545$
\item $n=1365$
\item $n=2907$
\item $n=132$
\item $n=462$
\end{enumerate}
\hfill
\hfill
\end{document}