\documentclass[11pt,twoside]{amsart}
\usepackage{amssymb, amsmath, enumerate, palatino, hyperref, xypic, mathabx, epsfig}

%%% Theorem environments %%%
\theoremstyle{remark}
\newtheorem{prob}{Problem}
\newtheorem{exc}[prob]{Exercise}
\newtheorem{chal}[prob]{Challenge}
%%%%%%%%%%%%%%%%%%

%%% Macros %%%%%%%%%%%
\newcommand{\CC}{\mathbb{C}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}

\newcommand{\normal}{\trianglelefteq}
\newcommand{\ul}{\underline}
\newcommand{\action}{\mathrel{\reflectbox{$\righttoleftarrow$}}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Inn}{\operatorname{Inn}}
\newcommand{\Out}{\operatorname{Out}}
\newcommand{\GSet}{G\text{-}\operatorname{Set}}
\newcommand{\mm}[4]{\begin{pmatrix}#1 &#2\\ #3 &#4\end{pmatrix}}
\newcommand{\Syl}{\operatorname{Syl}}
%%%%%%%%%%%%%%%%%%

\title{Math 332: Homework 5}
%\author{Your name here --- delete the % at the start of the line}
\date{2.III.15}

\begin{document}
\maketitle

\begin{exc}
Exhibit all Sylow 2-subgroups and Sylow 3-subgroups of $D_{12}$ and $S_3\times S_3$.
\end{exc}

\begin{exc}
Exhibit two distinct Sylow 2-subgroups of $S_5$ and an element of $S_5$ that conjugates one into the other.
\end{exc}

\begin{exc}
Exhibit all Sylow 3-subgroups of $SL_2(\FF_3)$.
\end{exc}

\begin{exc}
Prove that the subgroup of $SL_2(\FF_3)$ generated by $\mm{0}{-1}{1}{0}$ and $\mm{1}{1}{1}{-1}$ is the \emph{unique} Sylow 2-subgroup of $SL_2(\FF_3)$.  (You may use your result from Problem 7 of Homework 2.)
\end{exc}

\begin{prob}
Prove that a group of order 312 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order.
\end{prob}

\begin{prob}
Let $P\in \Syl_p(G)$ and assume $N\normal G$.  Use the second Sylow theorem to prove that $P\cap N$ is a Sylow $p$-subgroup of $N$.  Deduce that $PN/N$ is a Sylow $p$-subgroup of $G/N$.
\end{prob}

\begin{prob}
Prove that if $N$ is a normal subgroup of $G$, then $n_p(G/N)\le n_p(G)$.
\end{prob}

\begin{chal}
Prove that if there exists a chain of subgroups $G_1\le G_2\le G_3\le \cdots \le G$ such that $G = \bigcup_{i=1}^\infty G_i$ and each $G_i$ is simple, then $G$ is simple.
\end{chal}

\begin{chal}
Find all normal subgroups of $S_n$ for $n\ge 5$.
\end{chal}

\begin{chal}
Prove that $A_n$ does not have a proper subgroup of index $< n$ for all $n\ge 5$.
\end{chal}

\begin{chal}[Completely optional but possibly fun]
Use a computer to find each $n<10,000$ that is not a power of a prime and that has some prime divisor $p$ such that $n_p$ is not forced to be $1$ for all groups of order $n$ by the congruence condition of Sylow's theorem.  For each such $n$, give the prime factorization of $n$ and for each prime factor $p$ list the permissible values of $n_p$ (given the third part of Sylow's theorem).  Say something interesting about the statistics of these numbers and potential implications for the distribution of simple groups.
\end{chal}

\end{document}