\documentclass[12pt]{letter}
\usepackage{amssymb}
\textwidth 6.8in
\textheight 9.4in 
\topmargin -0.95in
\oddsidemargin -0.2in
\newcommand{\R}{{\mathbb R}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\K}{{\mathbb K}}

\newcommand{\vier}[4]{\left[ \begin{array}{ccc}
                   #1 &\;& #2 \\ #3 &\;& #4 \end{array} \right]}

\begin{document}
\normalsize
\baselineskip 0.47cm  %\thispagestyle{empty}

\begin{center}
  {\bf Test 1 for Honors Algebra III, Math 361 \\ }
  {\footnotesize \hfill October 12, 1998}\\
\end{center}
Instructions: The test is 50 minutes in length.




\begin{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Let $z$ be the complex number $z=1+i$. Find
  $a,b\in\R$ such that $z^{-1}=a+bi$.
  
  \vfill (10 pts) Prove by induction that
  $$
  1+2+\cdots+n=\frac{1}{2}n(n+1).
  $$

 
  \vfill
  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (10 pts) Find all groups of order $31$. Explain why your
  list is complete.
  
  \vfill (10 pts) Find all {\bf abelian} groups of order $8$.
  Explain why your list is complete.
  
  
  \vfill
  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (A) (10 pts) Let $p$ be a prime. Show that the nonzero elements
  of $\Z_p$ form a group under multiplication.
  
  
  \vfill (B) (10 pts) Assume $p$ is a prime, $a$ a positive integer.
  Prove with details that
  $$
  a^p\equiv a \mbox{ mod }p.
  $$

  
  \vfill

  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (20 pts) Prove in detail that a group of order 9 is
  abelian.

  
  \newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item (20 pts) Let
  $$
  H:= {\rm rowsp}_{\Z}\vier{8}{-3}{1}{4}.
  $$
  Compute the invariant factor decomposition of
  $\vier{8}{-3}{1}{4}$. Thereafter describe a finite abelian
  group which is isomorphic to ${\Z}^2/H$.



 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\end{enumerate}
\end{document}
