\documentclass[12pt]{article}

\usepackage[margin=0.5in,top=0.7in,dvips]{geometry}
\usepackage{fancyhdr,lastpage}
\usepackage{amsmath,amsthm,amsfonts}
\usepackage{tikz}
\usepackage{multirow}
\usepackage{substr}
\usepackage{cancel}

\DeclareMathOperator{\Aut}{Aut}

\newtheorem{prop}{Proposition}
\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{conj}{Conjecture}
\newtheorem{cor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}

\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{#1}}
\makeatother

% Include "solution" in your filename to make sure the right mode is
% used below.

\IfSubStringInString{\detokenize{solution}}{\jobname}{%
  % Solutions
  \newcommand\qfill{}
  \newcommand\qpage{}
  \newcommand\qonly[1]{}
  \newcommand\answer[1]{#1}
  \newcommand\question[1]{{\em #1}}
}
{%
  % Problems
  \newcommand\qfill{\vfill}
  \newcommand\qpage{\newpage}
  \newcommand\qonly[1]{#1}
  \newcommand\answer[1]{}
  \newcommand\question[1]{#1}
}

\pagestyle{fancy}
\lhead{MATH 521--01}
\chead{Problem Set \#3\answer{ solutions}}
\rhead{}
\lfoot{}
\cfoot{\answer{Page \thepage\ of \pageref{LastPage}}}
\rfoot{due Thursday, October 2, 2014}

\begin{document}
\begin{enumerate}
\item \question{Prove that $(1234)$ cannot be written as the product
    of 3-cycles.}

  \answer{}


\item \question{Fnd an element of $S_9$ with order 20, and prove that
    no element of $S_8$ has order 20.}

  \answer{}

\item \question{Prove that $S_4$ is not isomorphic to $D_{12}$.}

  \answer{}

\item \question{Recall that the \emph{centralizer} of an element $a$
    of a group, denoted, $C(a)$, is the subgroup consisting of all $b$
    such that $ab=ba$.

    Prove that for any $a,g\in G$, $C(a)$ is isomorphic to
    $C(gag^{-1})$.}

  \answer{}

\item \question{Describe $\Aut(D_5)$ as fully as possible: identify
    its order, its elements, and any multiplicative rules you can find.}

  \answer{}
\end{enumerate}
\end{document}