\documentclass{report} \usepackage{amsthm, amsmath, amssymb,verbatim,xspace, graphicx} \usepackage{pdfsync, listings, color,pgf,tikz} \usepackage[all,cmtip]{xy} \newcommand\R{{\mathbb R}} \newcommand\Q{{\mathbb Q}} \newcommand\N{{\mathbb N}} \newcommand\Z{{\mathbb Z}} \newcommand\dsp{\displaystyle} \newcommand\eP{\varphi} \newcommand\ns{\trianglelefteq} \newcommand\ldiv{\, \bigg | \,} \newcommand{\Mod[1]}{~( \operatorname{mod} \,#1)} \newcommand{\Aut}{\operatorname{Aut}} \newcommand\F{{\mathbb F}} \newcommand{\tcm}{\textcolor{magenta}} \setlength{\oddsidemargin}{-.25in} \setlength{\evensidemargin}{0in} \setlength{\textwidth}{7in} \setlength{\textheight}{9.5in} \setlength{\topmargin}{-.25in} \setlength{\headheight}{0in} \setlength{\headsep}{0in} \usepackage{textcomp} \newcommand{\tc}{\text{:}} % tc=tight colon, for use in math mode in Newick trees \begin{document} \noindent \bigskip \centerline{\sc \LARGE Sample Algebra Comprehensive Exam Problems} \medskip \centerline{Spring 2019} \vskip .5cm This sample exam is a composite of questions asked on recent MATH 631 exams. As a result the length of this exam is much longer (by a factor of at least 2) than a comprehensive exam, but the breadth of topics covered and the difficulty level are representative of possible test questions. \medskip \noindent {\bf Part I.} Short answer or easy computation. \begin{enumerate} \item Consider the cylic group $C_{4900} = \langle x \rangle$ of order $4900 = 2^2 \cdot 5^2 \cdot 7^2$. \begin{enumerate} \item Give the number of generators of $C_{4900}$. \item List explicitly the elements $x^a$, with $0 \le a \le 4899$, of order $10$. \tcm{\emph{Answer:}} $\vert x^a \vert = 10$ if $a = \underline{\hskip 12cm }$. \medskip (If it helps, you can simply give the prime factorizations of $a$. I am not interested in your ability to multiply integers.) \end{enumerate} \item Consider the cyclic groups $\Z/30\Z$ and $C_{18} = \langle x \rangle$ of orders $30$ and $18$ respectively, and suppose that \begin{align*} \varphi_a : \Z/30\Z &\to C_{18}\\ 1 &\mapsto x^a \end{align*} extends to a well-defined group homomorphism from $\Z/30\Z$ to $C_{18}$. \begin{enumerate} \item List the values of $a$ with $0 \le a \le 17$ for which this is true. (I.e. The map defines a well-defined group homomorphism.) \item Give a brief explanation why such a well-defined group homomorphism can not be surjective. \end{enumerate} \item Consider the symmetric group $G = S_{7}$ and let $\sigma = (1 \ 2 \ 3 \ 6 \ 5 \ 4 \ 7 )$ be a $7$-cycle. \begin{enumerate} \item Express $\sigma$ as the product of (not necessarily disjoint) transpositions. \item Compute the number of conjugates of $\sigma$ in $S_7$. \item Let $\tau$ be the $7$-cycle $(3 \ 7 \ 1 \ 4 \ 5 \ 6 \ 2)$. Give an element $\alpha$ that conjugates $\sigma$ to $\tau$, i.e. give $\alpha$ such that $\alpha \sigma {\alpha}^{-1} = \tau$. \item Noting that $S_7$ acts on itself by conjugation, explicitly use the Orbit-Stabilizer theorem to find the size of the stabilizer of $\sigma$ under this action and the elements of the Stabilizer subgroup of $S_7$. \tcm{The stabilizer of $\sigma$ in this context is better known as \underline{\hskip 3cm}. (Using appropriate notation in place of words here is fine.)} \item Noting that $\sigma \in A_7$, what is the size of the conjugacy class of $\sigma$ in $A_7$? Stated otherwise, how many conjugates in $A_7$ does $\sigma$ have? Briefly, state a result that justifies your answer. \tcm{\emph{Answer:} The number of conjugates of $\sigma$ in $A_7$ is \underline{\hskip 2cm}} \medskip \tcm{because ....} \end{enumerate} \item \begin{enumerate} \item Suppose that $A$ is an Abelian group of order $200 = 2^3 \cdot 5^2$. Give the isomorphism classes for $A$ in a table below. In the left hand column, give the elementary divisor decomposition and in the right hand column, give the invariant factor decomposition. {\bf Groups on the same row should be isomorphic.} You do not need to show your work. \item Give the number of non-isomorphic Abelian groups of order $400 = 2^4 \cdot 5^2$. \end{enumerate} \item Prove that there are no simple groups of order 56. \item Give the definition of a nilpotent element in a ring $R$. Then prove that the set of nilpotent elements in $M_2(\Q)$ is {\bf not} an ideal. \item Suppose $G$ is a non-cyclic group of order $205 = 5\cdot 41$. Give, with proof, the number of elements of order $5$ in $G$. \item Find {\bf ALL} solutions $x$ in the integers to the simultaneous congruences. \begin{align*} x \equiv \ &7 \mod{11}\\ x \equiv \ &2 \mod{5} \end{align*} \item Draw the lattice diagram of prime ideals for the polynomial ring $\Q[x]$. \emph{Note:} There are infinitely many prime ideals so you will need a way to indicate them all. \end{enumerate} \vskip 1cm \noindent {\bf Part II.} Theory \begin{enumerate} \item Suppose $G$ is a group with $H, K$ subgroups of $G$. Prove that if $H \le N_G(K)$, then $HK = \big \{ h k \mid h \in H, k \in K \big \}$ is a subgroup of $G$. \item Suppose that a finite group $G$ is of order 105, $\vert G \vert = 3 \cdot 5 \cdot 7$, and that $G$ has normal subgroups of order $3$, $5$ and $7$. Prove or disprove: $G$ is cyclic. \item Let $P$ be a $p$-group, $\vert P \vert = p^a > 1$ for $p$ a prime, and let $A$ be a nonempty finite set. Suppose that $P$ acts on $A$ and define \emph{the set of fixed points} of this action: $$A_0 = \big \{ a \in A \mid g \cdot a = a \text{ for every } g \in P \big \}.$$ Prove that $$ \vert A \vert \equiv \vert A_0 \vert \Mod[p]. $$ \item Let $\eP (n)$ denote the Euler $\varphi$-function. Prove that if $p$ is a prime and $n \in \Z^+$, then $$ n \ldiv \eP (p^n - 1). $$ (Hint: Compute the order of $\bar p$ in the appropriate group first.) \item Prove that if $G$ is a group of order $p^2$ for $p$ a prime, then $G$ is Abelian. \item Suppose $G$ is a finite group of order $\vert G \vert = 14,553 = 3^3 \cdot 7^2 \cdot 11$ and that $N$ is a normal subgroup of $G$ of order $\vert N \vert = 11$. Prove that $N \le Z(G)$. \item Suppose $G$ is a group, $H \le G$, and $\Aut (H)$ the group of automorphisms of $H$. \begin{enumerate} \item Using the First Isomorphism theorem, give a \tcm{\bf full} proof of the following statement. The quotient group $N_G(H) / C_G(H) \cong A \le \Aut(H)$. \item Suppose now that $P$ is a Sylow $p$-subgroup of $S_p$ for a prime $p$. Prove that $$ N_{S_p} (P) / C_{S_p} ( P ) \cong \Aut(P). $$ \end{enumerate} \item Let $G$ be a finite group of order 22. Prove that $G$ is cyclic or isomorphic to the dihedral group $D_{22}$. \item In a PID every nonzero element is a prime if, and only if, it is irreducible. \item Suppose $R$ is a commutative ring with 1 and for each $x \in R$, there is a positive integer $n > 1$ so that $x^n = x$. Prove that every nonzero prime ideal is maximal. \item Let $\F_{7}$ denote the finite field with 7 elements. \begin{enumerate} \item Explicitly construct a finite field with $343 = 7^3$ elements. Explain your work. \item The field you constructed in part (a) is a simple extension of $\F_{7}$ so let $\alpha$ be an element in some extension of $\F_7$ such that $\dsp \vert \F_{7} (\alpha) \vert = 343$. Find the inverse of the element $1 + \alpha \in \F_{7} (\alpha)$. \end{enumerate} \item Find, with brief justification, all ring homomorphisms from $\Z \to \Z/12Z$. \item Consider the ring of Gaussian integers $\Z[i]$. \begin{enumerate} \item Prove that if $\alpha = a + b i$ for $a, b \in \Z$ is a Gaussian integer with $N(\alpha) = p$ for $p$ a prime of $\Z$, then $\alpha$ is irreducible. \item List all the units of $\Z[i]$. \item Give an example of a prime number $p \in \Z$ such that $p$ is irreducible in $\Z[i]$. Justify your answer by stating an appropriate result. \end{enumerate} \item Let $D$ be a square-free integer, and consider the quadratic number field $\Q (\sqrt{D})$ and its subring of integers $\mathcal O$. Let $N: \Q(\sqrt{D}) \to \Z$ denote the field norm map which is multiplicative. The restriction of $N$ to the ring of integers $\mathcal O$ will also denoted by $N$. \begin{enumerate} \item Prove that an element $\alpha \in \mathcal O$ is a unit if, and only if, $N(\alpha) = \pm1$. \item When $D = -3$, the ring of integers is $\dsp \mathcal O = \Z + \Z \bigg(\frac{1 + \sqrt{-3}}{2}\bigg)$. Find a unit in $\mathcal O \smallsetminus \Z$. \item Let $D=-5$. Give, with proof, an example of an element $x = a + b \sqrt{-5}$ for $a, b \in Z$ such that $x$ is irreducible, but $x$ is not prime in $\Z [ \sqrt{-5}]$. \end{enumerate} \end{enumerate} \end{document}