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\begin{center}
Final Exam\\
Math 601\\
10 a.m. December 14 \\
Due: 12 noon December 15, 2000 \\
%Karen A. Chandler \\
\end{center}

{\bf Instructions:}
Please do questions 1 through 3.
You may obtain extra points by doing question~4.
To maximize credit, please 
present your answers in a clear and precise form.
Remember: it will be easier for me to understand your work if I can {\bf read}
it!

Good luck!



\begin{enumerate}

\item
Let $G$ be a group of order~231.  Let $H$ be a Sylow 11-subgroup, 
$K$ a Sylow 7-subgroup, and $L$ a Sylow 3-subgroup of $G$.
\begin{enumerate}
\item Show that $H$ and $K$ are both normal subgroups of $G$.
\item Show that $G$ has a cyclic subgroup of order~77.
\item Show that $G=HKL$.
\end{enumerate}

\item
Consider $\rats/\ints$ as a group under addition.
\begin{enumerate}
\item Prove that every finitely generated subgroup of $\rats/\ints$ is cyclic.
\item Prove that $\rats/\ints$ is not a finitely generated group.
\end{enumerate}


%\labitem{(30 points)}  For each of the following, either give an example or
%explain why no such example exists.
%\begin{enumerate}
%\item an element of an integral domain that is prime but not irreducible
%\item an element of an integral domain that is irreducible but not prime
%\end{enumerate}




%\labitem{(30 points) (10)} 
%\begin{enumerate}
%\item  Find polynomials $p(x)$ and $q(x)$ in $\rats[x]$ so that
%$(x^3-x-1)p(x)+(x^2-2)q(x)=1$.  
%
%{\it (Suggestion: use the Euclidean algorithm for polynomials.)}
%\labiitem{(10)} Prove that 
%$\rats[x]/(x^3-x-1)(x^2-2)$ is isomorphic to $\rats[x]/(x^3-x-1)\times \rats[x]/(x^2-2)$.
%
%\labiitem {(10)} Find a polynomial $f(x) \in \rats[x]$ such that
%$(x^3-x-1)$ divides $f(x)-1$ and $(x^2-2)$ divides $f(x)-x$.
%%Find a solution $f(x) \in \rats[x]$ to the system of congruences
%%$$ f(x) \equiv 1 ~(\mbox{mod}~ x-1),~ f(x) \equiv x ~(\mbox{mod}~ x^2-2).$$
%Describe the set of all such $f(x)$.
%%\labiitem {(14)} Find the set of all solutions $g(x) \in \rats[x]$  of the system of congruences 
%%$$ g(x) \equiv 1 ~(\mbox{mod}~ x-1),~ g(x) \equiv x ~(\mbox{mod}~ x^2-2),~g(x) \equiv 1 ~(\mbox{mod}~ x+1).$$
%
%${\it (Suggestion: use part (b) to express the system in terms of just two congruence equations.)}
%\end{enumerate}
%
\item
Let $R$ be a commutative ring with identity element $1 \ne 0$.  
Recall that an element $r \in R$ is {\it nilpotent} if there exists
$n \in $~{\bf N} such that $r^n=0$.
\begin{enumerate}
\item Show that the set of nilpotent elements of~$R$ is an ideal.
\item Show that if $r$ is nilpotent then $1-r$ is a unit.

%{\em (Suggestion: multiply it by $1+r+r^2+ \ldots r^{n-1}$.)}.
\item Suppose $a_0 \in R$ is a unit, and $a_1, \ldots, a_n$ are
nilpotent.  Let $f=a_0+a_1x+\ldots+a_n x^n \in R[x]$.
Show that $f$ is a unit in $R[x]$.
%

%{\em (Hint: is $a_1 x+\ldots+ a_n x^n$ nilpotent? )}

\end{enumerate}



%\labitem{(40 points)} Let $S=\Bbb C[x, y]$.
%Let $f \in S$, so we may write 
%$f = \sum_{1 \leq i+j \leq n} \alpha_{i,j} x^i y^j$, $\alpha_{i,j} \in \cx$.
%Let $p \in \cx^2$, and write $p=(a,b), a, b \in \cx$.
%Then we write $f(p)= \sum_{1 \leq i+j \leq n} \alpha_{i,j} a^i b^j$.
%\begin{enumerate}
%\item Let $X$ be a subset of $\cx^2$.  
%Let $$I(X)=\{f \in S: f(p)=0 \mbox{~for~all~} p \in X\}.$$
%Show that $I(X)$ is an ideal of $S$.
%\item Given an ideal $I \leq S$,
 %call $$V(I)=\{p \in \cx^2: f(p)=0 \mbox{~for~all~} f \in I\}.$$
%Show that $X \subset V(I(X))$.
%\item Recall that 
%$$\mbox{rad} (I) = \{ f \in S: f^n \in I \mbox{~for~some~} n \in \nats\}.$$
%Show that $I \subseteq \mbox{rad}(I) \subseteq I(V(I))$.
%\end{enumerate}

\labitem{(Bonus)}  Let ${\cal C}$ be a concrete category.
Suppose that for any two objects $A,B$ of ${\cal C}$,
there is an object $A*B$ of ${\cal C}$ and morphisms
$${\iota}_A:A \rightarrow A*B,~ {\iota}_B:B \rightarrow A*B$$
so that $A*B$, together with the morphisms 
${\iota}_A$ and ${\iota}_B$, is a coproduct
of $A$ and $B$.
\begin{enumerate}
\item Suppose that $A,~B,~C$ are three objects of ${\cal C}$.
Show that $(A*B)*C$ (with appropriate morphisms) is a coproduct of $A,~B,~C$.
\item Show that $A*(B*C)$ is a coproduct of $A,~B,~C$.
\item  Conclude that $(A*B)*C$ is isomorphic to $A*(B*C)$.
\item Did we need the hypothesis that ${\cal C}$ is a concrete category?

{\it (Suggestion: draw the diagrams.)}
\end{enumerate}


\end{enumerate}
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