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\section*{Project 2}
\subsection*{Due: November 9, 2007}
\textbf{Recommended length: 3-5 pages.}\\
\textbf{Format: typed.}
\vspace{1cm}
\noindent The goal of this assignment is to examine the behavior of in and out shuffles. Suppose that you have a deck of $2n$ cards. Let $i\in S_{2n}$ denote the permutation that gives an in-shuffle, and let $o\in S_{2n}$ denote the permutation that gives the out-shuffle. Let
$$B_n=\left\{\begin{array}{c} \text{$n\times n$ matrices with entries in $\{-1,0,1\}$}\\ \text{and exactly one nonzero entry in}\\ \text{every row and in every column}\end{array}\right\}.$$
For $b\in B_n$, let $\bar{b}\in W_n$ be the same matrix as $b$ but with $-1$'s replaced by $1$'s.
The goals of the project are to
\begin{enumerate}
\item[(1)] Prove the following two results.
\begin{theorem}
The group $\langle i,o\rangle$ is isomorphic to a subgroup of $B_n$.
\end{theorem}
\begin{corollary}
The order of $\langle i, o\rangle$ is less than or equal to $2^n n!$.
\end{corollary}
\item[(2)] Investigate the behavior of the three following functions.
$$\begin{array}{rcl}
\sigma:B_n&\longrightarrow& \{-1,1\}\\
b & \mapsto & \det(b)\end{array}\quad
\begin{array}{rcl}
\bar\sigma:B_n&\longrightarrow& \{-1,1\}\\
b & \mapsto & \det(\bar{b}),\end{array}\quad
\begin{array}{rcl}
\sigma\times\bar\sigma:B_n&\longrightarrow& \{-1,1\}\\
b & \mapsto & \det(b)\det(\bar{b}).\end{array}$$
In particular, show that each is a homomorphism, and describe their kernels.
\end{enumerate}
\noindent Note that this is a writing assignment, so the main focus should be on clearly communicating the ideas in the proof. I recommend looking at your favorite mathematics texts and trying to emulate their style. I also suggest you have another member of the class read through a draft before handing it in.
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