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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
\end{tabular}
\end{center}

\vskip 5mm
\centerline{{\bf Part I: Groups} (Choose at least two.)}
\vskip 3mm


\num
\item  
\num
\item  Prove that if a subgroup \(G\) of \(S_n\) contains an odd permutation, then \(|G|\) is even and exactly half of the elements of \(G\) are odd permutations.
\item  Show that for \(n \geq 5\), \(A_n\) is the only subgroup of index 2 in \(S_n\).  (You may use the fact that \(A_n\) is simple for \(n \geq 5\).)
\item  Give an example of a nonabelian group with more than one subgroup of index 2.
\mun

\item 
\num
\item  Prove that every group of order 105 has a nontrivial proper normal subgroup.
\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 composition.
\mun

\item  
Label each of the following statements as true or false.  Justify each answer with a proof or a counterexample.
\num
\item  If \(G\) is a group with \(|G| = n < \infty\) and \(k|n\), then \(G\) has a subgroup of order \(k\).
\item  If \(G\) is an abelian group with \(|G| = n < \infty\) and \(k|n\), then \(G\) has a subgroup of order \(k\).
\item If \(G\) is a group of order $p^k$ for $p$ prime, then \(G\) has an abelian normal subgroup of order greater than 1.
\item  If \(G\) is a group with \(|G| = n < \infty\), \(k|n\) and \(k\) is a power of a prime, then \(G\) has a subgroup of order \(k\).
\mun

\item  Given a group $G$, define the commutator subgroup to be the subgroup generated by commutators in $G$. That is, 
\[
[G, G] := \langle aba^{-1}b^{-1} : a, b \in G \rangle.
\]
\num
\item Show that $[G, G]$ is a normal subgroup of $G$.
\item Show that any group homomorphism $\varphi: G \rightarrow A$ with $A$ abelian, takes $[G, G]$ to the the trivial subgroup in $A$. 
\item For $G = D_8$ the dihedral group with 8 elements, compute the subgroup $[G, G]$ and the quotient $G/[G, G]$.
\item Find all group homomorphisms $\varphi: D_8 \rightarrow \bC^*$, where $\bC^*$ denotes the group of non-zero complex numbers under multiplication.
\mun

\item  
\num
\item  Let \(G\) and \(H\) be finite groups.  Prove that if \(|G|\) and \(|H|\) are relatively prime, then the only homomorphism \(\phi: G \rightarrow H\) is the trivial one (i.e. \(\phi(g) = e_H\) for all \(g \in G\), where \(e_H\) is the indentity of \(H\)).
\item  How many automorphisms does \(S_3\) have?  Justify your answer.
\mun

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


\item  
Let \(R = \ZZ[i]\) be the ring of Gaussian integers, where \(i^2 = -1\).
\num
\item  Prove that \(R\) is a unique factorization domain.  You may use without proof the fact that a Euclidean domain is a unique factorization domain.  
\item  Prove that 3 is irreducible in \(R\), but that 5 is not.
\item  Prove that the ideal \(2R\) is not maximal.  
\mun

\item  Let \(f(x) = x^6 - 1, g(x) = x^8 - 1\) in the ring \(\RR[x]\).
\num
\item  Give a complete factorization of \(f\) and \(g\) into irreducibles.
\item  Calculate \(\gcd(f,g)\) and express your answer as a linear combination of \(f\) and \(g\).
\item  For ideals \(A, B\) of a ring, we define the {\em sum} \mbox{\(A + B = \setbuild{a + b}{a \in A, b \in B}\)}  and {\em product} \(AB = \setbuild{\sum_{i=1}^n a_i b_i}{n \in \ZZ^+, a_i \in A, b_i \in B}\).  For each of the following ideals, find a monic polynomial that generates it.
\num
\item \((f) + (g)\)
\item \((f)(g)\)
\item \((f) \cap (g)\)
\mun
\mun

\item  
\num
\item  Prove that every finite integral domain is a field.
\item  Let \(D\) be an integral domain and let \(I\) be a prime ideal of \(D\) that has finite index.  Prove that \(I\) is a maximal ideal of \(D\).
\item  Give an example of an integral domain with a nonzero prime ideal that is not maximal.  
\mun


\item  
Let \(R\) be a commutative ring with identity \(1 \neq 0\).  Suppose that \(e,f \in R\) such that \(ef =0, e+f = 1\).
\num
\item  Prove that \(e^2 = e\) and \(f^2 = f\).
\item  Prove that the principal ideal \(eR\) generated by \(e\) is a ring with \(e\) as its identity.
\item  Prove that \(R \isom eR \times fR\) as rings.
\mun

\item  Construct a \(3 \times 3\) matrix having an eigenvalue 1 with corresponding eigenvector \vec{1\\1\\0}, an eigenvalue of \(-1\) with corresponding eigenvector \vec{1\\0\\1}, and an eigenvalue of 2 with corresponding eigenvector \vec{0\\1\\1}.  Is this matrix unique, or could there be others with this property?  Justify your answer.  
\mun

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