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\begin{center} \underline {SECTION I} \end{center}
\begin{enumerate}
\item Let $G$ be a finite group and let $p$ be a prime. Suppose that every proper subgroup of $G$ of order divisible by $p$ is a $p$-group. Show that every two distinct Sylow $p$-subgroups of $G$ intersect trivially. ({\it Hint}: \ Consider the normalizer of a maximal intersection of Sylow $p$-subgroups.)
%2
\item Let $G$ be a finite simple group.
\begin{enumerate}
\item If $G$ has a proper subgroup of index less than or equal to 9, show that $G$ has no elements of order 21.
%b
\item If $|G|$ = 504, show that $G$ has no elements of order 21.
%c
\item If $|G|$ = 504, find the number of Sylow 7-subgroups of $G$. (Justify your answer.)
\end{enumerate}
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\begin{center} \underline {SECTION II} \end{center}
%3
\item Let $E/F$ be a finite dimensional field extension, and let $G$ be the group of automorphisms of $E$ which fix each element of $F$. Suppose that for some $u \in E$ the elements $\sigma (u)$ (as $\sigma$ ranges over $G$) form a basis for $E$ over $F$.
\begin{enumerate}
%a
\item Show that if an element $v$ of $E$ is fixed by each $\sigma$ in $G$, then $v$ is in $F$.
%b
\item Show that $E = F[u]$.
\end{enumerate}
%4
\item Let $E/F$ be a field extension. Give a complete proof of the fact that $E$ is an algebraic extension of $F$ if and only if each subring of $E$ that contains $F$ is a field.
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\begin{center} \underline{SECTION III} \end{center}
%5
\item Let $R$ be a primitive ring in which $a(ab - ba) = (ab - ba) a$ \ for all $a$ \ and $b$\ in $R$. \ Show that $R$ is a division ring.
%6
\item A ring $R$ is called {\it prime} if whenever $I$ and $J$ are ideals of $R$ with $IJ$ = 0, then $I$ = 0 or $J$ = 0. ($IJ$ is the additive subgroup of $R$ generated by the set of products $xy$ with $x \in I$ and $y \in J$.)
\begin{enumerate}
%a
\item Show that if a prime ring has no nonzero nilpotent elements then it has no nonzero zero divisors. (Any properties of prime rings that you use other
than the definition must be verified.)
%b
\item Show that a primitive ring is prime.
\end{enumerate}
%7
\item Let $M$ be a finitely generated unital right $R$-module over the right noetherian ring $R$. \ Show that $M$ is right noetherian.
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\begin{center}{\bf DEPARTMENT OF MATHEMATICS \\
Ph.D. Comprehensive Exam \\
Algebra \\
September 1996} \end{center}
\vspace{1in}
\begin{center} {Instructions \\
\bigskip
Do four problems. Do at least \\
one problem from each section.\\
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Please make sure that you give complete \\
solutions to each problem that you do.\\
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If you attempt more than four problems \\
please indicate which ones you wish to have graded.} \end{center}
\vspace{1in}
\begin{center} {Policy on Misprints} \end{center}
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The Ph.D. Comprehensive Exam Committee tries to proofread the exams as carefully as possible. Nevertheless, the exam may contain misprints. If you are convinced a problem has been stated incorrectly, mention this to the proctor and indicate your interpretation in your solution. In such cases do not interpret the problem in such a way that it becomes trivial.
\end{enumerate}
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