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\name{}
\class{Math 171 - Abstract Algebra I Section \uline{\;\;}}
\assignment{HW \#20}
\duedate{10/28/10}

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\begin{problem} 1. Let $G$ be a group of order $p^{a}$ and let $N$ be a non-trivial normal subgroup of $G$. Show that $N \cap Z(G)$ is a non-trivial subgroup of $G$, where $Z(G)$ is the center of $G$. (In particular, $Z(G)$ is non-trivial.)
\end{problem}

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\begin{problem} 2. \begin{itemize} \item[(a.)] Show that a Sylow $p$-subgroup of a group $G$ is normal if and only if it is the unique Sylow $p$-subgroup of $G$.
\item[(b.)] Show that a group of order 56 has a normal Sylow $p$-subgroup for some prime $p$ dividing its order. (Hint: Counting elements might be helpful in one case.)
\end{itemize}
\end{problem}

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\begin{problem}[3. (University of Wisconsin Algebra Qualifying Exam, August '03 Problem 1)] Let $G$ be a group of order 504.

\begin{itemize}
\item[(a.)] Show that $G$ cannot be isomorphic to a subgroup of the alternating group $A_{7}$.
\item[(b.)] If $G$ is simple, determine the number of Sylow $3$-subgroups of $G$.
\end{itemize}

Note: For this problem, you will need the $n!$ theorem (Problem 2 from HW 20) and the fact that $A_7$ itself is simple. Also, if $H$ is a subgroup of $S_{7}$, $H \cap A_{7}$ is often quite interesting.... This was one problem of five on this exam, and solving it completely earned a little less than one-third of the score needed to pass the exam.)

\end{problem}

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