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\title{General Mathematics and CPS II}
\author{Exercise 14}
\date{March 28, 2014}
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\begin{enumerate}

\item Kac ring paper, Exercise 14.

\item Fix $N \in \mathbb N$.  Let $S$ denote the $N \times N$
\emph{circular shift matrix}
\[
  S =   
  \begin{pmatrix}
    0 & 0 & \cdots & 0 & 1 \\
    1 & 0 & 0 & \cdots & 0 \\
    0 & 1 & 0 &  \cdots & 0 \\
    \vdots  & \ddots & \ddots & \ddots & \vdots \\
    0  & \cdots & 0 & 1 & 0 
  \end{pmatrix}
\]

Further, let $M$ denote the diagonal matrix
\[
  M = \begin{pmatrix}
  m_1 & 0 &  \\
  0 & m_2 & \ddots \\
  & \ddots & \ddots & \ddots \\
  &  & \ddots & \ddots & 0 \\
  & & & 0 & m_N
  \end{pmatrix}
\]
where $\lvert m_1 \rvert = \dots = \lvert m_N \rvert = 1$.

Show that $\{ (SM)^i \colon i \in \mathbb Z \}$ is a group.
What is its order?

\item There is a correspondence of the construction in Questions~3
with the Kac ring model.  Explain!  Further, there is a correspondence
of this construction with glide reflections.  Explain!
\end{enumerate}



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