Preliminary Exam - Fall 1989

Problem 1   Let $A$ be a finite abelian group, and $m$ the maximum of the orders of the elements of $A$. Put $S=\{a \in A \;\vert\; \vert a\vert = m\}$. Prove that $A$ is generated by $S$.

Problem 2   Let $f:[0,1]\to\mbox{$\mathbb{R}^{}$}$ be a real valued continuously differentiable function with $f(0)=0$. Suppose also that there is a constant $M>0$ such that, for $0\leq x\leq 1$,

$$0 \leq f'(x) \leq M f(x).$$

Prove that $f(x)=0$ for $0\leq x \leq 1$.

Problem 3   Let $A$ be a real, upper-triangular, $n\times n$ matrix that commutes with its transpose. Prove that $A$ is diagonal.

Problem 4   Let $f(z)$ be analytic for $\vert z\vert<1$ and suppose that

$$\vert f(z)\vert \leq \frac{1}{1 - \vert z\vert}.$$

Show that $\vert f'(0)\vert\leq 4$.

Problem 5   Let $G$ be a group, $G'$ its commutator subgroup, and $N$ a normal subgroup of $G$. Suppose that $N$ is cyclic. Prove that $gn = ng$ for all $g\in G'$ and all $n\in N$.

Problem 6   Let $X \subset \mbox{$\mathbb{R}^{n}$}$ be a closed set and r a fixed positive real number. Let $Y = \{y \in \mbox{$\mathbb{R}^{n}$} \,\vert\, \vert x-y\vert=r \; for \,\, some \; x \in X\}$. Show that $Y$ is closed.

Problem 7   Let $A$ and $B$ be diagonalizable linear transformations of $\mathbb{R}^{n}$ into itself such that $AB = BA$. Let $E$ be an eigenspace of $A$. Prove that the restriction of $B$ to $E$ is diagonalizable.

Problem 8   Evaluate the integral

$$I = \int_0^{\infty}\frac{\log x}{1+x^2}\,dx\, .$$

Problem 9   Let $\mbox{\bf {F}}$ be a field, $\mbox{\bf {F}}[x]$ the polynomial ring in one variable over $\mbox{\bf {F}}$, and $R$ a subring of $\mbox{\bf {F}}[x]$ with $\mbox{\bf {F}} \subset R$. Prove that there exists a finite set $\{f_1,f_2,\ldots,f_n\}$ of elements of $\mbox{\bf {F}}[x]$ such that $R = \mbox{\bf {F}}[f_1,f_2,\ldots,f_n]$.

Problem 10   Let $\alpha$ be a number in $(0,1)$. Prove that any sequence $(x_{n})$ of real numbers satisfying the recurrence relation

$$x_{n+1} = \alpha x_{n} + (1 - \alpha )x_{n-1}$$

has a limit, and find an expression for the limit in terms of $\alpha$, $x_0$ and $x_1$.

Problem 11   Let $\varphi$ be Euler's totient function; so if $n$ is a positive _Euler>totient function integer, then $\varphi(n)$ is the number of integers $m$ for which $1\leq m \leq n$ and $\gcd \{n,m\} = 1$. Let $a$ and $k$ be two integers, with $a>1$, $k>0$. Prove that $k$ divides $\varphi(a^k-1)$.

Problem 12   Let $f(z)$ be analytic in the annulus $\Omega = \{1<\vert z\vert<2\}$. Assume that $f$ has no zeros in $\Omega$. Show that there exists an integer $n$ and an analytic function $g$ in $\Omega$ such that, for all $z \in \Omega$, $f(z)=z^ne^{g(z)}$.

Problem 13   Let ${A}$ be an $n\times n$ real matrix, ${A}^t$ its transpose. Show that ${A}^t{A}$ and ${A}^t$ have the same range. In other words, given $y$, show that the equation $y = {A}^t{A}x$ has a solution $x$ if and only if the equation $y = {A}^tz$ has a solution $z$.

Problem 14   Let $X \subset\mbox{$\mathbb{R}^{n}$}$ be compact and let $f:X \to \mbox{$\mathbb{R}^{ }$}$ be continuous. Given $\varepsilon > 0$, show there is an $M$ such that for all $x, y\in X$,

$$\vert f(x)-f(y)\vert \leq M\vert x-y\vert + \varepsilon.$$

Problem 15   Let $G_n$ be the free group on $n$ generators. Show that $G_2$ and $G_3$ are not isomorphic.

Problem 16   Prove that the polynomial

$$p(z)=z^{47}-z^{23}+2z^{11}-z^5+4z^2+1$$

has at least one root in the disc $\vert z\vert<1$.

Problem 17   Let $V$ be a vector space of finite-dimension $n$ over a field of characteristic $0$. Prove that $V$ is not the union of finitely many subspaces of dimension $n-1$.

Problem 18   Let the function $f$ from $[0,1]$ to $[0,1]$ have the following properties:

• $f$ is of class $C^1$;

• $f(0)=f(1)=0$;

• $f'$ is nonincreasing (i.e., $f$ is concave).

Prove that the arclength of the graph of $f$ does not exceed $3$.