\documentclass[12pt]{article}
\usepackage{amsmath,amsfonts}

\input{ExamMacros.tex}


\prof{Chris Bendel and Peter Cholak} \class{Math 222 - Quiz 10}
\quizdate{Wednesday, April 7} \pagestyle{empty}

\begin{document}


\heading

Be sure to carefully write up your answers.  It is suggested that you
first write out a draft of your proposed questions and then carefully
rewrite that draft to get your final version.  You do \emph{not} have
to write the answers on this sheet of paper.\\

\begin{problem}
  Determine which of the following sets form a group.  If not,
explain why.

(a) The permutations of odd {\em order} in $S_4$ under permutation
multiplication.

(b) The positive real numbers under multiplication.

(c) The postive real numbers under addition.

(d)  The set $3\mathbb{Z} \equiv \{0, \pm 3, \pm 6, \pm 9, \pm 12,
\dots \} \subset \mathbb{Z}$ under addition.
\end{problem}

\begin{problem}
 Write out the multiplication table for the dihedral group $D_3$
obtained from an equilateral triangle.  Orient the triangle so that
one of the points is facing upward and define the ``flip'' to be
sideways as done in class for the square and $D_4$.
\end{problem}

\begin{problem}
  Let $n \geq 3$ be a non-negative integer.  How many elements of
order two does the dihedral group $D_n$ have? {\em Hint:} The answer
may depend on whether $n$ is even or odd.
\end{problem}

\end{document}
             