\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 12}
\quizdate{Friday, April 23} \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}
  Consider the subgroup $A_4 \subset S_4$ of even permutations.
For each of the following integers, determine whether $A_4$ has a
subgroup of that order.  If so, explicitly exhibit a such a subgroup
and if not, explain why not.

a) 2

b) 5

c) 3

d) 4

e) 8
\end{problem}

\begin{problem}
  (Section 9.4, Problem 42.)  Suppose that $H$ is a subgroup of $G$
  and $g$ is an element of $G$.  Prove that the set $$gHg^{-1} =
  \{ghg^{-1}: h \in H\}$$
  is a subgroup of $G$ and is isomorphic to
  $H$.
\end{problem}

%\begin{problem}
%  Compute the following centralizers.

%a) $Z_2((\zz_5,+))$

%b) $Z_{F}(D_4)$

%c) $Z_{(12)}(S_3)$
%\end{problem}

%\begin{problem} Up to isomorphism, how many groups are there of
%order 83?  Why?
%\end{problem}

\begin{problem}
  (Section 9.5, Problem 25.) Prove that a group has exactly 2 subgroups
  iff it is isomorphic to $(\mathbb{Z}_p,+)$ for some prime $p$.
\end{problem}
\end{document}
             