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\noindent M.A. COMPREHENSIVE EXAM \hfill ALGEBRA \hfill JULY 17, 1999\\
\noindent {\bf Please give complete proofs. Do {\it two} problems from each of the three parts. If you do three problems in one of the parts please indicate which two problems you want graded.}
\begin{center} PART I. \end{center}
\begin{enumerate}
\item Let $G$ be a finite abelian group and let $z$ be the product of all of the elements in $G$. Prove that $z^2=1$. Give an example of $G\neq 1$ where $z=1$ and another example where $z\neq 1$.
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\item Let $n\geq 3$ and let $H$ be the subgroup of the symmetric group $S_n$ which is generated by the set of 3-cycles. Show that $H$ is $A_n$, the alternating group on $n$ letters.
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\item Let $G$ be a finite $p$-group and let $H$ be a normal subgroup of $G$ of order $p$. Show that $H$ is contained in the center of $G$.
\bigskip
\begin{center} PART II. \end{center}
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\item Prove that ${\Bbb Z}[x]$, the polynomial ring over the integers ${\Bbb Z}$, is not a principal ideal domain.
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\item Let $R$ be a commutative ring with an identity element. Under addition $R$ is an abelian group. Suppose that each subgroup of this group is an ideal of $R$. Show that the ring $R$ is isomorphic to the ring of integers ${\Bbb Z}$, or to the integers modulo $n$, for some integer $n$.
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\item Let $R$ be a commutative ring with an identity element and let $a$ be an element of $R$. Suppose that $M$ is an ideal of $R$ with the following two properties:
\begin{enumerate}
\item $a^n \not \in M$ for $n=1,2, \cdots, \ .$
\item If $K$ is an ideal of $R$ which contains $M$ but $K\neq M$, then $a^n\in K$ for some $n$. \ Prove that $M$ is a prime ideal of $R$; that is, if $x,y\in R$ and $xy\in M$ then $x\in M$ or $y\in M$.
\end{enumerate}
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\begin{center} PART III. \end{center}
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\item Let $A$ be a real symmetric $m\times m$ matrix and suppose that $A^n=I$ for some $n\geq 1$. \ Show that $A^2=I$.
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\item Let $T$ and $S$ be $n\times n$ matrices with entries from the field $K$. Suppose that $N$ is the null-space of $S$ (i.e., $N$ is the set of $n\times 1$ column vectors annihilated by $S$). \ Show that $TN\subseteq N$.
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\item Let $T$ be an $n\times n$ matrix over the field $K$. \ Suppose that $p(x)$ is a nonzero polynomial of least degree with $p(T)=0$. Show that $T$ is invertible if and only if the constant term of $p(x)$ is $\neq 0$.
\end{enumerate}
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