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

\def \rr{\mathbb{R}}
\def \zz{\mathbb{Z}}
\def \cc{\mathbb{C}}
\def \qq{\mathbb{Q}}

\input{ExamMacros.tex}


\prof{Chris Bendel and Peter Cholak} \class{Math 222 - Quiz 11}
\quizdate{Monday, April 19} \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}
  Which of the following are isomorphisms?  Explain why or why not.

(a) $f: (\rr > 0, \cdot) \to (\rr > 0, \cdot)$ by $f(x) = x^2$.

(b)  $f: (\rr, +) \to (\rr, +)$ by $f(x) = x^3$.

(c) Let $G = \Big{\{}\begin{pmatrix}
a & a\\
a & a\\
\end{pmatrix}: a \in \rr\Big{\}}$ under matrix addition.  $f:
(\rr,+)
\to G$ by $f(a) = \begin{pmatrix}
a & a\\
a & a\\
\end{pmatrix}$.

(d)  $f: (\zz,+) \to (\zz,+)$ by $f(x) = 2x$.
\end{problem}

\begin{problem}
 Consider the subgroup $H = \{Id, (1234), (13)(24), (1432)\}$ of
$S_4$.  Up to isomorphism, there are only two groups of order 4:
$(\zz_4,+)$ and $K$.  Which is $H$?  Why?
\end{problem}

\begin{problem}
  Find a subgroup of $S_4$ of order 4 which is isomorphic to the
other one, i.e.\ the one that wasn't the answer to the previous
problem.
\end{problem}

\end{document}
             