\documentclass[11pt]{article}
\usepackage{amsmath,amsthm,amsfonts,amssymb,amscd}
\usepackage{graphicx, epsfig}
\usepackage{multicol}
\usepackage{boxedminipage}
\usepackage{subfigure}
\usepackage{euler,palatino,array}
\usepackage[colorlinks=true, pdfstartview=FitV, linkcolor=blue, citecolor=blue, urlcolor=blue]{hyperref}
\usepackage{wrapfig}
\usepackage{bbold}


\setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.5in} 
\setlength{\parindent}{0in}
\newcounter{prob}
\setcounter{prob}{0}
\newcommand{\problem}{\stepcounter{prob}\medskip{\bf Presentation problem \theprob.}\ }
\newcommand{\answer}{\medskip{\bf Answer to Question \theprob.}\ }
\newcommand{\solution}{\medskip{\bf Solution to Question \theprob.}\ }



% ***** Some useful abreviations *****
\newcommand{\abs}[1]{\left|#1\right|}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\K}{\mathbb{K}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\B}{\mathscr{B}}
\newcommand{\ra}{\rightarrow}
\newcommand{\ds}{\displaystyle}
\newcommand{\bnum}{\begin{enumerate}} %Use with \item to create numbered lists
\newcommand{\enum}{\end{enumerate}}
\newcommand{\babc}{\bnum\renewcommand{\labelenumi}{(\alph{enumi})}}%Use with \item for abc lists
\newcommand{\eabc}{\end{enumerate}}
\newcommand{\bijk}{\bnum\renewcommand{\labelenmi}{\roman{enumi}.}}%Roman numeral lists
\newcommand{\eijk}{\end{enumerate}}

\renewcommand{\ss}{\smallskip}
\newcommand{\ms}{\medskip}
\newcommand{\bs}{\bigskip}




\pagestyle{myheadings} \markright{Math 3340 (Spring 2021) \hspace{2in}
(HW6)}

\begin{document}


%% PREAMBLE

\thispagestyle{empty}

\begin{minipage}[t]{1.1in}
{\vspace{-0.2in}
  \includegraphics[width=1.1in]{cornell-logo.jpg}}
\end{minipage}
\hfill
\begin{minipage}[t]{3in}
%\vspace{.05in}
  {\large Homework \# 6 $\quad$ Math 3340 Spring 2021

    \medskip
    
    Due at the end of the day Friday, April 2, at midnight.
  }

\end{minipage}
\hfill
\begin{minipage}[t]{3.4in}
\end{minipage}

\vskip 0.25in

%\vspace*{-66mm}


Please submit your completed homework via gradescope on canvas.
I encourage you to work with your classmates on this homework (except for the journal entries!)
When you submit your work, please {\bf list your collaborators}.
(Your grade will not be affected.)
Even if you work in a group, you should write up 
your solutions {\bf yourself}! 
You should include all computational details, and proofs should be carefully 
written with full details.
As always, please write {\bf neatly and legibly} (feel free to use \LaTeX\ to write up your solutions, if you wish!).

\vskip 4mm

%% END PREAMBLE

%%   \noindent {\bf Reading.} {\em The Secret to Raising Smart Kids} by Carol Dweck, in Scientific American.  You can find it
%% by Google, or in the files section on our canvas site (it is also linked to from the syllabus).

%% \vskip 0.1in

\noindent {\bf Journal entry.} There is no journal entry this week.

\vskip 0.1in

\noindent {\bf Exercises.} 

\vskip 0.1in

\begin{enumerate}

\item Show that a group action $G \times S \to S$ is equivalent to a group homomorphism
  $G \to Sym(S)$ (that is, given $G$ and $S$, find a 1-1 correspondence between the set
  of group actions $G \times S \to S$, and the set of group homomorphisms $G \to Sym(S)$.)
  
\item Suppose that $G$ is a finite group of order $n$, and $H$ is a proper subgroup of index $m$.
  \babc
\item Show that if $G$ is simple, then $G$ is isomorphic to a subgroup of $S_m$.
  (Hint: Use one of the problems from the first prelim).
  \item Show that if $n$ does not divide $m!$, then $G$ is not simple.
  \eabc

\item How many elements of order 7 can a a group of order 168 have?  (Note, there is
  a simple group of order 168.  We will meet it later!).

\item Suppose that $n > 1$ is an integer which is not prime.
  \babc
\item Given such an $n$, show that if for some prime $p$ dividing $n$, $1$ is the only divisor of $n$
  which is equivalent to $1$
  mod $p$, then any group $G$ of order $n$ is not simple.

  \medskip\noindent
  A computer search (or hand search) shows that the only numbers $n \le 60$ for which there doesn't exist such a prime $p$,
  are the numbers $\{ 12, 24, 30, 36, 48, 56, 60 \}$ (You do not need to do this search).
  
\item Find an odd $n$ which does not satisfy the hypothesis on $n$
  (i.e. find an odd $n$ such that for each prime $p$ dividing $n$,
  there is a divisor $>1$ of $n$, which is equivalent to $1$ mod $p$).
  \eabc

\pagebreak  
\item   There is a simple group of order 60 (we have not yet proved this, but it is $A_5$).
  In this problem, we show that there are no non-Abelian simple groups of order smaller than $60$.
From the last problem, it suffices to consider
  groups of order in the set $\{ 12, 24, 30, 36, 48, 56 \}$.
\babc  
  \item 
  Prove that there are no simple groups of order 12.
\item Prove that there are no simple groups of order 24.
\item Prove that there are no simple groups of order 30.
\item Prove that there are no simple groups of order 36.
\item Prove that there are no simple groups of order 48.
\item Prove that there are no simple groups of order 56. (note, this probably needs a different
  kind of proof than the others).
  \eabc

\item Here is another way to prove that there exists a Sylow $p$-subgroup.
  Let $G$ be a group of order $n = p^\alpha m$, where $p$ is prime, and $p$ does not divide $m$.
  Let $S$ be the set of all {\it subsets} of $G$ of order $p^\alpha$.
  In this problem you may use without proof the following fact: The number of subsets of such size, $p^\alpha m \choose p^\alpha$, (where $p$ does not divide $m$),
  is not divisible by $p$.
  \babc
\item Define a group action of $G$ on $S$ by $g * B := gB = \{ gb \mid b \in B \}$,
  where $B \subset G$.  Show that this is a group action.
\item Show that there is a set $B \in S$ whose orbit size is not divisible by $p$.
\item Let $H \le G$ be the stabilizer of this set $B$.
  Show that $H$ has at least $p^\alpha$ elements.
\item Show that actually, $|H| \le |B|$ and therefore that $H$ is a Sylow $p$-subgroup of $G$.
  (one way might be to start with a $b \in B$, and use that to create a function $f: H \to B$,
  given by $f(g) = g * b$.)
  \eabc

\end{enumerate}
\end{document}

