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\centerline{{\bf Part II: Rings and Linear Algebra} (Choose at least two.)}
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\markright{\tiny \copyright 20\examyear\ by California State University.  Unauthorized distribution of this material will result in civil and criminal prosecution.}

{\bf \semester\ 20\examyear\  -- Algebra Comprehensive Exam} \hfill {\mbox{\hskip 5mm Name: \rule{1.1in}{.005 in}}}

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Choose six problems total, including at least two from Part I and two from Part II.  Enter the numbers of the problems you want graded here:

\begin{center}\begin{tabular}{|l|c|c|c|c|c|c|l|} \hline
Problems & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & \makebox[.5cm] & Total\\ \hline
Scores & & & & & & & \\ \hline
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\centerline{{\bf Part I: Groups} (Choose at least two.)}
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\num

\item  \num
\item  Let \(G\) be a group with normal subgroups \(H\) and \(K\) such that \mbox{\(H \cap K = \{1\}\)}.  Prove that \(H \times K\) is isomorphic to \(HK\).
\item  Give an example to show that the result of part (a) is false if only one of the subgroups is normal.
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\item  \num
\item  Prove that a group of order $48$ has a normal subgroup of either order 16 or order 8.
\item  Classify all abelian groups of order $576=2^6\cdot 3^2$ up to isomorphism.
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\item  Show that if \(p < q < r\) are prime numbers and \(G\) is a group of order \(pqr\), then at least one of the Sylow subgroups of \(G\) is normal.  

\item  Let \(G\) be a group acting on a set \(A\).  
\num
\item  Prove that for any \(a \in A\), the orbit of \(a\) has order \(|G:G_a|\), where \(G_a\) is the stabilizer of \(a\) in \(G\).  
\item  Use this to prove that for any \(g \in G\), the order of the conjugacy class of \(g\) is \(|G:C_G(g)|\), where \(C_G(g)\) is the centralizer of \(g\) in \(G\).
\item  Let \(\sigma = \begin{pmatrix}1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 5 & 4 & 1 & 2 & 3 & 7 & 6\end{pmatrix} \in S_7\).  Determine the sizes of the conjugacy class of \sig\ and the centralizer \(C_{S_7}(\sig)\).  
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\item  Let \(G\) be a group.  Define the {\em commutator subgroup} of \(G\) to be the subgroup \(G'\) generated by all elements of the form \(aba\inv b\inv\), where \(a,b \in G\).
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\item  Prove that \(G'\) is a normal subgroup of \(G\).
\item  Prove that \(G/G'\) is abelian.
\item  Prove or give a counterexample to the statement that if \(\phi: G \rightarrow G\) is an automorphism, then \(\phi(G') \subseteq G'\).
\item  Find the commutator subgroup of \(D_n\), the dihedral group of order \(2n\).
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\item  \num  
\item  Let \(f(x) \in \ZZ[x]\) be a monic polynomial of degree \(\geq 1\).  The {\it Rational Root Theorem} states that if \(a \in \QQ\) with \(f(a) = 0\), then \(a \in \ZZ\).  Prove this theorem.
\item  In the ring \(\ZZ[x]\), let \(A\) be the principal ideal generated by \(x-1\) and let \(B\) be the principal ideal generated by \(x+1\).
\num
\item  Show that \(A \cap B\) is a principal ideal.
\item  Show that \(A + B\) is a maximal ideal that is not principal.  
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\item    
\num
\item  Show that \(R = \ZZ\bpoint{\sqrt{-2}}\) is a Euclidean domain under the usual norm, \(N\point{x + y\sqrt{-2}} = x^2 + 2y^2\).  In other words, show that given \(a, b \in R\) with \(b \neq 0\), there exist \(q, r \in R\) such that \(a = bq + r\) and \(N(r) < N(b)\).  (You may use without proof the fact that \(N(ab) = N(a)N(b)\).)
\item  The result in (a) is not true if \(-2\) is replaced by \(-3\).  Circle the line of your work for (a) that would fail if we replaced \(-2\) by \(-3\).  
\item  Show carefully that \(3+\sqrt2\) is irreducible in \(R = \ZZ\bpoint{\sqrt2}\).
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\item  \num
\item  Prove that every Euclidean domain is a PID.
\item  Prove that in a PID, every ascending chain of ideals
\[
I_0 \subseteq I_1 \subseteq I_2 \subseteq \cdots
\]
stabilizes; that is, there is a value \(n\) for which \(I_n = I_{n+1} = \cdots\).  (Hint:  Show that the union of all ideals in the chain is an ideal.)
\item  Give a proof or a counterexample to the following statement.  Let \(R\) be a PID.  If \(r \in R\) is irreducible, then \((r)\) is a maximal ideal.  
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\item  
Let \(R\) be an integral domain.  Label each of the following statements as true or false.  Justify each answer with a proof or counterexample.
\num
\item  For any two nonzero proper ideals \(I, J\) of \(R\), \(I \cap J \neq \{0\}\).
\item  Every nonzero prime ideal of \(R\) is maximal.
\item  Every prime element of \(R\) is irreducible.  
\item  If \(R\) is finite, then \(R\) is a field.
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\item  Let \(V\) be a finite-dimensional vector space.  Let \(T: V \rightarrow V\) be a linear transformation with \(T \circ T = T\).  
\num
\item  What are the possible eigenvalues for \(T\)?
\item  Let \(v \in V\) and set \(w_0 = (I - T)(v)\) and \(w_1 = T(v)\) (where \(I\) is the identity map).  Show that \(T\point{w_0} = 0, T\point{w_1} = w_1\), and \(v = w_0 + w_1\).
\item  Show that \(W = \{v: T(v) = v\}\) is a subspace of \(V\) and that \mbox{\(\ker T \cap W = \{0\}\)}.  
\item  Show that there exists a basis \ppoint{u_1, u_2, \dots, u_n} of \(V\) and an integer \(k \leq n\) such that 
\[
T\point{a_1 u_1 + a_2 u_2 + \cdots + a_n u_n} = a_1 u_1 + \cdots + a_ku_k.
\]
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