\documentclass[10pt]{article}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage[left=1.5in,right=1.5in,bottom=1.5in,top=1in]{geometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Begin user defined commands
\newcommand{\map}[1]{\xrightarrow{#1}}
\newcommand{\define}{\stackrel{\mathrm{def}}{=}}
\newcommand{\Z}{\mathbb Z}
\newcommand{\Q}{\mathbb Q}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\F}{\mathbb F}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\Aut}{\mathrm{Aut}}
\newcommand{\End}{\mathrm{End}}
\newcommand{\iso}{\cong}
\newcommand{\mil}{\varprojlim}
\newcommand{\dlim}{\varinjlim}
\newcommand{\ord}{\mathrm{ord}}
\newcommand{\action}{\boxempty}
\newcommand{\Stab}{\mathrm{Stab}}
\newcommand{\normal}{\lhd}
\newcommand{\lamron}{\rhd}
\newcommand{\ab}{\mathrm{ab}}
\newcommand{\semi}{\rtimes}
\newcommand{\imes}{\ltimes}
\newcommand{\alg}{\mathrm{alg}}
% End user defined commands
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\input xy
%\xyoption{all}
%\CompileMatrices
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% These establish different environments for stating Theorems, Lemmas, Remarks, etc.
\newtheorem{Thm}{Theorem}
\newtheorem{Prop}[Thm]{Proposition}
\newtheorem{Lem}[Thm]{Lemma}
\newtheorem{Cor}[Thm]{Corollary}
\theoremstyle{definition}
\newtheorem{exercise}[Thm]{Exercise}
\renewcommand{\labelenumi}{(\alph{enumi})}
% End environments
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Now we're ready to start
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\author{}
\title{Algebra I \\Problem Set \#2}
\date{}
\maketitle
\pagestyle{empty} % These lines turn off the automatic page numbering
\thispagestyle{empty}
\begin{exercise}
Suppose $G$ is a finite group with the property that every Sylow subgroup is normal. Prove that
\[
G \iso P_1\times \cdots \times P_r
\]
where $P_1,\ldots, P_r$ are the Sylow subgroups of $G$.
[In particular, a finite abelian group is the product of its $p$-Sylows]
\end{exercise}
\begin{exercise}
Let $P$ be a $p$-Sylow subgroup of a group $G$, and let $H$ be any subgroup
of $G$. Show that there is a $g\in G$ such that $g P g^{-1}\cap H$ is a
a $p$-Sylow subgroup of $H$, and that every $p$-Sylow subgroup of
$H$ is of this form for some $g\in G$.
\end{exercise}
\begin{exercise}
Show that there are no simple groups of order
\begin{enumerate}
\item
$p^2q$ for primes $p