# 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 $S = \{ a \in A \mid \left| a \right| = m \}$ . Prove that A is generated by S.

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

$$
0 \leqslant f ^ { \prime } ( x ) \leqslant M f ( x ) .
$$

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

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 $| z | < 1$ and suppose that

$$
| f ( z ) | \leqslant { \frac { 1 } { 1 - | z | } } .
$$

Show that $| f ^ { \prime } ( 0 ) | \leqslant 4$

Problem 5 Let G be a group, G0 its commutator subgroup, and N a normal subgroup of G. Suppose that N is cyclic. Prove that $g n = n g$ for all $g \in G ^ { \prime }$ and all $n \in N$

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

Problem 7 Let A and B be diagonalizable linear transformations of Rn into itself such that $A B = B A$ Let E be an eigenspace of A. Prove that the restriction of B to E is diagonalizable.

Problem 8 Evaluate the integral

$$
I = \int _ { 0 } ^ { \infty } { \frac { \log x } { 1 + x ^ { 2 } } } d x .
$$

Problem 9 Let F be a field, $\mathbf { F } [ x ]$ the polynomial ring in one variable over F, and R a subring of ${ \bf F } [ x ]$ with $\mathbf { F } \subset R$ . Prove that there exists a finite set $\{ f _ { 1 } , f _ { 2 } , \ldots , f _ { n } \}$ of elements of ${ \bf F } [ x ]$ such that $R = \mathbf { F } [ f _ { 1 } , f _ { 2 } , \ldots , f _ { n } ]$

Problem 10 Let α 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 integer, then $\varphi ( n )$ is the number of integers m for which $1 \leqslant$ m $\leqslant$ 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 $\Omega = \{ 1 < | z | < 2 \}$ . Assume that f has no zeros in Ω. Show that there exists an integer n and an analytic function g in Ω such that, for all $z \in \Omega , f ( z ) = z ^ { n } e ^ { g ( z ) }$

Problem 13 Let A be an n×n real matrix, At its transpose. Show that AtA 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 $y = A ^ { t } z$ has a solution z .

Problem 14 Let $X \subset \mathbb { R } ^ { n }$ be compact and let $f : X \to \mathbb { R }$ be continuous. Given $\varepsilon > 0$ , show there is an M such that for all $x , y \in X$ 7

$$
| f ( x ) - f ( y ) | \leqslant M | x - y | + \varepsilon .
$$

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

$$
p ( z ) = z ^ { 4 7 } - z ^ { 2 3 } + 2 z ^ { 1 1 } - z ^ { 5 } + 4 z ^ { 2 } + 1
$$

has at least one root in the disc $| z | < 1$

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 $C ^ { 1 }$ ,

$f ( 0 ) = f ( 1 ) = 0 ,$

$f ^ { \prime }$ is nonincreasing (i.e., f is concave).

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