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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:

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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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\item  Let \((G,*)\) be a group with operation \(*\) and identity \(e\). Let \(a\) be a fixed element of \(G\). Define a binary operation \(\triangle\) on \(G\) as \(g\triangle h = g*a*h\) for all \(g,h\in G\). 
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\item Prove that \((G,\triangle)\) is a group. In particular, make sure to identify the group identity under $\triangle$.
\item Show that \((G,*)\isom (G,\triangle)\) by exhibiting a group isomorphism.
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\item \num
\item Determine all finite groups that have no nontrivial proper subgroups. Justify your answer.
\item Determine all finite groups that have {\em exactly} one nontrivial proper subgroup. Justify your answer.
\item Give an example of a finite group \(G\) such that the number of subgroups of \(G\) is strictly greater than \(|G|\), the order of \(G\).
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\item \num
\item Show that a group of order 28 must contain a normal subgroup of order 7.
\item Show that a group of order 28 must contain a normal subgroup of order 14.
\item Give an example of a group of order 28 that does not contain a normal subgroup of order 4. Justify your answer.
\item How many isomorphism classes of abelian groups of order 700 are there? For each one give its invariant factor decomposition.
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\item 
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\item Let \(G\) be a finite group with center \(Z(G)\). Prove that if \(G/Z(G)\) is cyclic, then \(G\) is abelian.
\item Let \(p\) be a prime number, and let \(G\) be a finite {\em \(p\)-group}; that is, a group of order \(p^n\) for some integer \(n\). 
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\item Prove that \(Z(G)\) is non-trivial.
\item Prove that if \(|G|=p^2\), then \(G\) is abelian.
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\item Show that if \(G\) is a group of order 135, then \(|Z(G)| > 1\).

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\item Let \(R\) be an integral domain. 
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\item Prove: If \(R\) is finite, then \(R\) is a field.
\item Prove: If \(I\) is a prime ideal of \(R\) such that \apoint{R:I} is finite, then \(I\) is a maximal ideal.
\item Give an example to show that part (b) is false without the assumption that \apoint{R:I} is finite.
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\item Let \(R = \ZZ\bpoint{\sqrt{-5}}\), and let \(I=\point{3, 2+\sqrt{-5}}\), the ideal of \(R\) generated by 3 and \(2+\sqrt{-5}\).
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\item Determine the units of \(R\).
\item Show that 3 is irreducible in \(R\).
\item Show that \(I\) is not a principal ideal.
\item Determine a set of coset representatives for \(R/I\).
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\item Let \(R\) be a commutative ring with identity. An element \(x\in R\) is {\em nilpotent} if \(x^n=0\) for some positive integer \(n\).
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\item Prove that the set \(N(R)\) of nilpotent elements of \(R\) is an ideal of \(R\). (You may use the Binomial Theorem without proof.)
\item Prove that if \(x\in R\) is nilpotent, then \(x\) is in every prime ideal of \(R\).
\item Determine the nilpotent elements of the ring \(\ZZ/72\ZZ\).
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\item Let \(n \geq 2\) and consider the ring \(\MM_n(\RR)\) of all \(n\times n\) matrices with real entries.
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\item Exhibit a proper nontrivial left ideal of \(\MM_n(\RR)\).
\item Prove that \(\MM_n(\RR)\) has no proper nontrivial two-sided ideals. 
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\item Let \(n\) be a positive integer. Recall the standard inner product \(\cdot\) on \(\RR^n\), given as follows: If \(\vx = \vec{x_1 \\ x_2\\ \vdots\\x_n}, \vy = \vec{y_1\\ y_2\\ \vdots\\ y_n}\), then \(\vx \cdot \vy = \ds \sum_{i=1}^n x_i y_i\). A set of vectors \(\vu_1, \vu_2, \dots, \vu_m\) in \(\RR^n\) is {\em orthonormal} if \(\vu_1\cdot\vu_j = \begin{cases} 1, & i=j \\ 0, & i\neq j \end{cases}\).
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\item For fixed \(\vy\in\RR^n\), prove that \(\vx \mapsto \vx\cdot\vy\) gives a linear transformation from \(\RR^n\) to \RR.
\item Prove that an orthonormal set in \(\RR^n\) is linearly independent.
\item If \(m\) is another positive integer and \(M\) is an \(m\times n\) matrix with orthonormal columns, prove that \((M\vx)\cdot(M\vy) = \vx\cdot \vy\) for all \(\vx, \vy \in \RR^n\).
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