\documentclass[amstex]{article} \usepackage{amsmath, amssymb, amstext, amsopn, amsxtra, graphicx, color, calligra, cancel, hyperref, url, ulem, pdfpages, permute, polynom, ifthen, float, epstopdf, linsys, marvosym, mathtools, rotating, soul, wrapfig, xifthen} \newcommand{\num}{\begin{enumerate}} \newcommand{\mun}{\end{enumerate}} \newcommand{\tem}{\begin{itemize}} \newcommand{\met}{\end{itemize}} \newcommand{\sig}{\ensuremath{\sigma}} \newcommand{\inv}{^{-1}} \newcommand{\field}[1]{\ensuremath{\mathbb #1}} \newcommand{\QQ}{\field Q} \newcommand{\ZZ}{\field Z} \newcommand{\point}[1]{\ensuremath{\left(#1\right)}} \newcommand{\ppoint}[1]{\ensuremath{\left\{#1\right\}}} \newcommand{\bpoint}[1]{\ensuremath{\left[#1\right]}} \newcommand{\examyear}{25} \newcommand{\semester}{Fall} \setlength{\parindent}{0 in} \setlength{\topmargin}{-0.5in} \setlength{\textheight}{9in} \begin{document} \newcommand{\ringpage}{ \pagebreak \centerline{{\bf Part II: Rings and Linear Algebra} (Choose at least two.)} \vskip 3mm} \pagestyle{empty} \pagenumbering{gobble} \pagestyle{myheadings} \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 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. \mun \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. \mun \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 \(C_{S_7}(\sig)\). \mun \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 \(a,b \in G\). \num \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 \(2n\). \mun \ringpage \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. \mun \mun \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 \(R = \ZZ\bpoint{\sqrt2}\). \mun \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. \mun \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. \mun \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 \[ T\point{a_1 u_1 + a_2 u_2 + \cdots + a_n u_n} = a_1 u_1 + \cdots + a_ku_k. \] \mun \mun \end{document}