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%Copied (mostly) from Fall 2019 (groups) and Fall 2021 (rings and linear algebra).

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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}}

\vskip 3mm

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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\vskip 5mm
\centerline{{\bf Part I: Groups} (Choose at least two.)}
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\num
\item  Let \(H\) and \(K\) be finite groups.  Let \(G = H \times K\).
\num
\item  If \(h \in H\) has order \(m\) and \(k \in K\) has order \(n\), what is the order of \point{h,k} in \(G\)?  Justify your answer.
\item  How many elements of order 20 are there in the group \linebreak \mbox{\((\ZZ/2\ZZ) \times (\ZZ/4\ZZ) \times (\ZZ/10\ZZ)\)}?

\mun

\item  In each item below you are given a group \(G\) with a subgroup \(H\).  Determine if \(H\) is a normal subgroup of \(G\).  Justify your answers.
\num
\item  \(G\) is a finite group with a unique element \(b\) of order 2; \(H = \gen b\).
\item  \(G = S_4\); \(H = \gen{(123)}\).
\item  \(G = D_{12}\), the dihedral group of order 12; \(H\) is a Sylow 2-subgroup of \(G\).  
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\item  
\num
\item  Prove that there are no simple groups of order $105$.
\item  How many isomorphism classes of abelian groups of order $360$ are there?  For each one give both its invariant factor decomposition and its elementary divisor decomposition.
\mun

\item  Let \(G\) be a group, and let \(H\) be a {\em normal} subgroup of \(G\), and let \(K\) be any subgroup of \(G\). 
\num
\item  Prove that \(HK = \setbuild{hk}{h \in H, k \in K}\) is a subgroup of \(G\).
\item  Now suppose further that \(H\) has index \(p\), where \(p\) is a prime number.  Prove that either \(K\) is a subgroup of \(H\) or \([K:K\cap H] = p\).
\mun

\item  Find all automorphisms of the group \(\ZZ \times \ZZ/n\ZZ\).  Your answer should include the following:
\num
\item  Describe each automorphism; that is, say what \(\pi(a,b)\) is for each automorphism \(\pi\) and each \((a,b) \in \ZZ \times \ZZ/n\ZZ\).
\item  Prove that each function you describe really is injective and surjective.
\item  Prove that there are no other automorphisms.
\item  State how many automorphisms there are in all.
\mun

\pagebreak
\centerline{{\bf Part II: Rings and Linear Algebra} (Choose at least two.)}
\vskip 3mm

\item  \num
\item   Let \(R\) be a commutative ring with identity.  Prove that \(R\) is a field if and only if the only ideals of \(R\) are \(\{0\}\) and \(R\).
\item  Give an example of a ring that has exactly three ideals.
\item  Show that \(\MM_2(\RR)\), the ring of \(2\times 2\) matrices with entries from the real numbers, has no nontrivial proper two-sided ideals but is not a division ring.
\mun

\item  Let \(\phi:R \rightarrow S\) be a homomorphism of rings, \(I\) an ideal of \(R\), \(J\) an ideal of \(S\).   
\num
\item  Prove that \(\phi\inv(J)\) is an ideal of \(R\).
\item  Prove that if \(\phi\) is {\em surjective}, then \(\phi(I)\) is an ideal of \(S\).
\item  Give an example to show that the previous part need not be true if \(\phi\) is not surjective.
\mun

\item  Let \(R = \ZZ\bpoint{\sqrt{-3}}\). 
\num
\item  Prove that the elements \(1 + \sqrt{-3}, 1 - \sqrt{-3}\), and 2 are all irreducible in $R$.
\item  Prove that \(R\) is not a unique factorization domain.
\item  Prove that \(R = \ZZ\bpoint{\sqrt{-3}}\) is not isomorphic to \(S = \ZZ\bpoint{\sqrt{-2}}\).
\mun

\item  
\num  
\item  Let \(R\) be a PID.  If \(I\) is a nonzero prime ideal of \(R\), prove that \(I\) is maximal.
\item  Give an example of an integral domain \(R\) with a nonzero prime ideal that is not maximal.  Justify your answer.
\item  In \(\ZZ_2[x]\), is the ideal generated by \(x^3+1\) a prime ideal?  Justify your answer.
\item  Consider the quotient ring 
\[
\ZZ_2[x]\modbig{\point{x^3+1}} = \setbuild{a + bX + cX^2}{a,b,c \in \ZZ_2},
\]
where \(X\) is the residue of \(x\) modulo \point{x^3+1}.
\num
\item  List all the units of \(R\).
\item  List all zero-divisors of \(R\).
\item  List all ideals of \(R\).  Which of them are prime?
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\item  \num
\item  Let \(A\) be an \(n \times n\) matrix with real entries having eigenvalues \(\lambda_1 \neq \lambda_2\) and associated eigenvectors \(\vv_1, \vv_2\) respectively.  Show that \(\vv_1\) and \(\vv_2\) are linearly independent.  
\renewcommand{\vec}[1]{\ensuremath{\begin{bmatrix}#1\end{bmatrix}}}
\item  Let \(A\) be the \(5 \times 5\) matrix whose entries are all 1, that is,
\[
A = \vec{1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1\\ 1 & 1 & 1 & 1 & 1}.
\]
\num

\item  Determine the eigenvalues for \(A\).
\item  Determine bases for the eigenspaces of \(A\).
\item  Is \(A\) diagonalizable?  If so, give a diagonal matrix similar to \(A\).
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