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\newcommand{\Z}{\mathbb{Z}}
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\begin{document}
\prof{George McNinch}
\date{\today}
\examname{Exam 3}
\class{Math 222}
\heading

This examination contains 4 problems on 5 sheets of paper.  Show all
your work.  I doubt that calculators will help much.

\vspace{1cm}
\begin{centerline}
  {\bf \sc Points}
\end{centerline}

\begin{center}
\begin{tabular}[t]{ccc}
Question & Possible\quad &\ Earned  \\ \hline\hline
  1      &   25     &       \\[.5pt] 
  2      &   25     &       \\[.5pt]
\end{tabular}
\quad
\begin{tabular}[t]{ccc}
Question & Possible\quad &\ Earned  \\ \hline\hline
  3      &   25     &       \\[.5pt]
  4      &   25     &       \\[.5pt]
Total    & 100\mbox{\  \ } & \\
\end{tabular}
\end{center}


\begin{problem}
  Consider the group $S_7$, the symmetric group on $\{1,2,\dots,7\}$.
  Let
  \begin{equation*}
    \sigma = 
    \begin{pmatrix}
      1 & 2 & 3 & 4 & 5 & 6 & 7 \\
      7 & 3 & 2 & 1 & 4 & 5 & 6
    \end{pmatrix} \in S_7
  \end{equation*}
  \begin{subproblem}
    Write $\sigma$ in cycle notation.
  \end{subproblem}
  \vspace{2.5cm}
  \begin{subproblem}
    Compute the order of $\sigma$.
  \end{subproblem}
  \vspace{2.5cm}
  \begin{subproblem}
    Decide whether $\sigma$ lies in $A_7$, the alternating group. Be
    sure to explain carefully your reasoning.
  \end{subproblem}
\end{problem}

\newpage
\begin{problem}
  You are given that the group of units of $\Z_{11}$ is cyclic and is
  generated by, for example, the number 6 which therefore has order 10
  (you may check this fact if you like, but this is not required.)
  
  Recall we showed in class that $\Z_{11}[i]$ is a field, where $i^2 =
  -1 = 10$. There are $11^2 -1 = 120$ units in $Z_{11}[i]$.
  \begin{subproblem}
    Let $\alpha = 4+3i \in \Z_{11}[i]$. Show that $\alpha$ has order
    $60$ in $\Z_{11}[i]$. (Hint: what is the order of $\alpha^{6}$?)
  \end{subproblem}
  \vspace{5cm}
  \begin{subproblem}
    Find a solution $\beta$ in $\Z_{11}[i]$ to the equation $X^2 =
    \alpha$. Deduce that $\beta$ has order 120 and hence that the unit
    group of $\Z_{11}[i]$ is cyclic. (See hint below.)
    \vspace{5cm}

    (Hint: if $\beta = a + bi$, then $\beta^2 = (a^2 - b^2) + 2abi,$
    so you need to find $a,b \in \Z_{11}$ so that $a^2 - b^2 = 4$ and
    $2ab=3$. The following table may be helpful -- you fill in the $a^2$
    column.
    \begin{equation*}
      \begin{matrix}
        b & a = \frac{3}{2b} & & b^2 & a^2 \\
        \hline
        1 & 7  && 1 & \\
        2 & 9  && 4 & \\
        3 & 6  && 9 & \\ 
        4 & 10 && 5 & \\
        5 & 8  && 3 & \\
      \end{matrix}
      \qquad
      \begin{matrix}
        b & a = \frac{3}{2b} & & b^2 & a^2 \\
        \hline
        6 & 3  && 3 & \\
        7 & 1  && 5 & \\
        8 & 5  && 9 & \\
        9 & 2  && 4 & \\
        10& 4  && 1 & \\
      \end{matrix}
    \end{equation*}
    Now find $\beta$.)
  \end{subproblem}
\newpage
\begin{subproblem}
  Find a unit $u \in \Z_{11}[i]$ or order 15. (You may leave your
  representation of $u$ in any form you like).
\end{subproblem}
\end{problem}

\newpage

\begin{problem}
  Consider the $d$ cycle $\sigma_d = (1, 2, 3, 4, \dots, d)$ in $S_n$
  for $1 \le d \le n$. In particular, $\sigma_1 = 1$, the identity
  permutation.
  \begin{subproblem}
    Fix $2 \le d \le n$. Find a transposition $\tau=(i,j) \in S_n$ so
    that $\sigma_d \tau = \sigma_{d-1}$.
  \end{subproblem}
  \vspace{6cm}
  \begin{subproblem}
    Using (a), prove by induction that $\text{sign}(\sigma_d) =
    (-1)^{d-1}.$
  \end{subproblem}
    \vspace{6cm}
  \begin{subproblem}
    What is the condition on $d$ so that $\sigma_d \in A_n$, the
    alternating group?
  \end{subproblem}
\end{problem}

\newpage

\begin{problem}
  Consider the group $D_{8}$, the group of symmetries of a 8-gon.
  Let $r = (123\dots 8)$ be the clockwise rotation of the 8-gon;
  $r$ generates the group $R_{8} < D_{8}$.
  
  Let $f_1 = (28)(37)(46)$ denote the ``flip'' which leaves
  the vertices 1 and 5 of the 8-gon fixed.
  \vspace{1cm}
  \begin{subproblem}
    Prove that $r^4 f_1 = f_1 r^4$.
  \end{subproblem}
  \vspace{7cm}
  \begin{subproblem}
    Let $H = \{ 1, r^4, f_1, r^4 f_1 \}$. Prove that $H$ is a subgroup
    of $G$ (of order 4, of course).
  \end{subproblem}
  \vspace{7cm}
  \begin{subproblem}
    Prove that $H$ is not a cyclic group, but that $H$ \emph{is}
    Abelian (i.e commutative).
  \end{subproblem}
\end{problem}

%\newpage
%\begin{problem}
%  Let $R = \Z[\sqrt{-3}] = \{ a + b \sqrt{-3} : a,b \in \Z\}$.
%%  \begin{subproblem}
%%    Prove that $3$ is not prime in $R$. Write $3$ as a product of
%%    primes in $R$. Remember to justify why the factors you give are
%%    actually prime.
%%  \end{subproblem}
%%  \vspace{6cm}
%  \begin{subproblem}
%    If $\alpha= a + b \sqrt{-3} \not = 0$, then $\alpha$ has an
%    inverse in the field $\Q(\sqrt{-3})$; say $\alpha^{-1} = c + d
%    \sqrt{-3}$ for $c,d \in \Q$. Give a formula for $c$ and $d$ in
%    terms of $a$ and $b$.
%  \end{subproblem}
%  \vspace{10cm}
%  \begin{subproblem}
%    Use (a) to give a guess for the inverse matrix $(M_\alpha)^{-1}$
%    of
%\begin{equation*}
%  M_{\alpha}=
%  \begin{bmatrix}
%    a & -3b \\
%    b & a
%  \end{bmatrix}
%\end{equation*}
%Verify that your guess works by computing the matrix product.
%\end{subproblem}
%\end{problem}

\end{document}

