# Preliminary Exam - Fall 1991

Problem 1 Prove that every finite group of order at least 3 has a nontrivial automorphism.

Problem 2 Let f be a continuous function from R to R such that $\left| f ( x ) - \right.$ $f ( y ) | \geqslant | x - y |$ for all x and y. Prove that the range of f is all of R. Note: See also Problem ??.

Problem 3 Evaluate the integral

$$
I = { \frac { 1 } { 2 \pi i } } \int _ { C } { \frac { z ^ { n - 1 } } { 3 z ^ { n } - 1 } } d z ,
$$

where n is a positive integer, and C is the circle $| z | = 1$ , with counterclockwise orientation.

Problem 4 1. Prove that any real n×n matrix M can be written as $M =$ $A + S + c I$ , where A is antisymmetric, S is symmetric, c is a scalar, I is the identity matrix, and tr $S = 0$

2. Prove that with the above notation,

$$
\mathrm { t r } ( M ^ { 2 } ) = \mathrm { t r } ( A ^ { 2 } ) + \mathrm { t r } ( S ^ { 2 } ) + \frac { 1 } { n } ( \mathrm { t r } M ) ^ { 2 } .
$$

Problem 5 Let f be an infinitely differentiable function from R to R. Suppose that, for some positive integer n,

$$
f ( 1 ) = f ( 0 ) = f ^ { \prime } ( 0 ) = f ^ { \prime \prime } ( 0 ) = \cdot \cdot \cdot = f ^ { ( n ) } ( 0 ) = 0 .
$$

Prove that $f ^ { ( n + 1 ) } ( x ) = 0$ for some x in (0, 1).

Problem 6 Let the function f be analytic in the disc $| z | < 1$ of the complex plane. Assume that there is a positive constant M such that

$$
\int _ { 0 } ^ { 2 \pi } | f ^ { \prime } ( r e ^ { i \theta } ) | d \theta \leqslant M , \qquad ( 0 \leqslant r < 1 ) .
$$

Prove that

$$
\int _ { [ 0 , 1 ) } | f ( x ) | d x < \infty .
$$

Problem 7 Consider the vector differential equation

$$
\frac { d x ( t ) } { d t } = A ( t ) x ( t )
$$

where A is a smooth $n \times n$ function on R. Assume A has the property that $\langle \mathrm { A } ( t ) y , y \rangle \leqslant c \| y \| ^ { 2 }$ for all y in $\mathbb { R } ^ { n }$ and all t, where c is a fixed real number. Prove that any solution $x ( t )$ of the equation satisfies $\| x ( t ) \| \leqslant e ^ { c t } \| x ( 0 ) \|$ for all $t > 0$

Problem 8 Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , . . .$ . be positive numbers.

1. Prove that $\textstyle \sum a _ { n } < \infty$ implies $\sum { \sqrt { a _ { n } a _ { n + 1 } } } < \infty$

2. Prove that the converse of the above statement is false.

Problem 9 Let G be a group of order $2 p ,$ where p is an odd prime. Assume that G has a normal subgroup of order 2. Prove that G is cyclic.

Problem 10 Let F be a finite field of order p. Compute the order of $S L _ { 3 } ( \mathbf { F } )$ the group of $\mathcal { B } \times \mathcal { B }$ matrices over F of determinant 1.

Problem 11 Let X and Y be metric spaces and f a continuous map of X into Y . Let $K _ { 1 } , K _ { 2 } , \ldots$ be nonempty compact subsets of X such $t h a t K _ { n + 1 } \subset$ $K _ { n }$ for all $n ,$ and let $K = \bigcap K _ { n }$ . Prove that $f ( K ) = \bigcap f ( K _ { n } )$

Problem 12 Let p be a nonconstant complex polynomial whose zeros are all in the half-plane $\Im z > 0$

1. Prove that $\Im ( p ^ { \prime } / p ) > 0$ on the real axis.

2. Find a relation between deg p and

$$
\int _ { - \infty } ^ { \infty } { \mathfrak { I } } { \frac { p ^ { \prime } ( x ) } { p ( x ) } } d x .
$$

Problem 13 Let $A = ( a _ { i j } ) _ { i , j = 1 } ^ { n }$ be a real $n \times n$ matrix with nonnegative entries such that

$$
\sum _ { j = 1 } ^ { n } a _ { i j } = 1 \qquad ( 1 \leqslant i \leqslant n ) .
$$

Prove that no eigenvalue of A has absolute value greater than 1.

Problem 14 Let B denote the unit ball of R3, $B = \{ r \in \mathbb { R } ^ { 3 } \mid \| r \| \leqslant 1 \}$ . Let $J = \left( J _ { 1 } , J _ { 2 } , J _ { 3 } \right)$ be a smooth vector field on $\mathbb { R } ^ { 3 }$ that vanishes outside of B and satisfies $\begin{array} { r } { \nabla \cdot \vec { J } = 0 } \end{array}$

1. For f a smooth, scalar-valued function defined on a neighborhood of B, prove that

$$
\int _ { B } \left( \nabla f \right) \cdot \vec { J } d x d y d z = 0 .
$$

2. Prove that

$$
\int _ { B } J _ { 1 } d x d y d z = 0 .
$$

Problem 15 Let I be the ideal in the ring $\mathbb { Z } [ x ]$ generated by $x - 7$ and 15. Prove that the quotient ring $\mathbb { Z } [ x ] / { \mathfrak { I } }$ is isomorphic to $\mathbb { Z } _ { 1 5 }$

Problem 16 Let $M _ { n \times n }$ be the space of real n×n matrices. Regard it as a metric space with the distance function

$$
d ( A , B ) = \sum _ { i , j = 1 } ^ { n } \left| a _ { i j } - b _ { i j } \right| \qquad ( A = ( a _ { i j } ) , B = ( b _ { i j } ) ) .
$$

Prove that the set of nilpotent matrices in $M _ { n \times n }$ is a closed set.

Problem 17 Let f be a C1 function from the interval (−1, 1) into $\mathbb { R } ^ { 2 }$ such that $f ( 0 ) = 0$ and $f ^ { \prime } ( 0 ) \neq 0$ Prove that there is a number ε in $( 0 , 1 )$ such that $\| f ( t ) \|$ is an increasing function of t on $( 0 , \varepsilon )$

Problem 18 Let the function f be analytic in the entire complex plane and satisfy the inequality $| f ( z ) | \leqslant | \Re z | ^ { - 1 / 2 }$ off the imaginary axis. Prove that f is constant.