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%TCIDATA{Created=Fri Dec 07 10:07:38 2001}
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\begin{document}
\title{Conjectures}
\date{December 2001}
\author{}
\maketitle
\begin{enumerate}
\item[1.] (BJT conjecture) Every subgroup $H$ of a finite group $G$ which
has index 2 in $G$ is a normal subgroup of $G.$
\begin{enumerate}
\item[ ] (CEK corollary) The subgroup of all rotations of $D_{n}$ is a
normal subgroup of $D_{n}.$
\end{enumerate}
\item[2.] (9-conjecture) The commutator subgroup of any group is normal.
\item[3.] (7-conjecture) The center of any group is normal.
\item[4.] (6-conjecture) Let $n$ be odd, and let $H$ be the commutator
subgroup of $D_{n}.$ Then $D_{n}/H\cong Z_{2}.$
\item[5.] (DJKT conjecture) Let $n$ be even and let $K$ be the commutator
subgroup of $D_{n}.$ Then $D_{n}/K\cong Z_{2}\times Z_{2}.$
\item[6.] (Kevin's conjecture) Let $n$ be even and let $C=\{r_{0},r_{n/2}\}$
be the subgroup of $D_{n}$ consisting of the identity and the 180$^{\circ }$
rotation. Then $C$ is normal in $D_{n}$ and $D_{n}/C\cong D_{n/2}.$
\end{enumerate}
\medskip
Students who contributed the conjectures:
\begin{itemize}
\item BJT: Brian, Jon, Tege
\item CEK: Charlie, Erika, Kevin
\item DJKT: Dale, Jon, Kevin, Tege
\item 9: Brian, Charlie, Dale, Jean-Marc, Jon, Kevin, Krista, Otto, Tege
\item 7: Brian, Charlie, Dale, Jean-Marc, Kevin, Krista, Otto
\item 6: Erica, Jon, Kevin, Krista, Otto, Tege
\end{itemize}
\end{document}