# Preliminary Exam - Spring 1997

Problem 1 For which values of the exponents $\alpha , \beta$ does the following series converge?

$$
\sum _ { n = 3 } ^ { \infty } { \frac { 1 } { n ^ { \alpha } ( \log n ) ^ { \beta } } } .
$$

Problem 2 Let M be a metric space with metric d. Let C be a nonempty closed subset of M. Define $f : M \to \mathbb { R } \ b y$

$$
f ( x ) = \operatorname* { i n f } \{ d ( x , y ) | y \in C \} .
$$

Show that f is continuous, and that $f ( x ) = 0$ if and only if $x \in C$

Problem 3 Suppose that $f ( x )$ is continuous, nonnegative for $x \geqslant 0$ , with $\textstyle \int _ { 0 } ^ { \infty } f ( x ) d x < \infty$ . Prove that

$$
\operatorname* { l i m } _ { n \to \infty } \int _ { 0 } ^ { n } { \frac { x f ( x ) } { n } } d x = 0 .
$$

Problem 4 Let f and g be two entire functions such that, for $a l l z \in \mathbb { C }$ $\Re f ( z ) \ : \leqslant \ : k \Re g ( z )$ for some real constant k (independent $o f z )$ Show that there are constants a, b such that

$$
f ( z ) = a g ( z ) + b .
$$

Problem 5 Prove that

$$
\int _ { - \infty } ^ { \infty } \frac { e ^ { - ( t - i \gamma ) ^ { 2 } / 2 } } { \sqrt { 2 \pi } } d t
$$

is independent of the real parameter γ.

Problem 6 Suppose that X is a topological space and V is a finite-dimensional subspace of the vector space of continuous real valued functions on X. Prove that there exist a basis $\{ f _ { 1 } , \ldots , f _ { n } \}$ for V and points $x _ { 1 } , \ldots , x _ { n }$ in X such that $f _ { i } ( x _ { j } ) = \delta _ { i j }$

Problem 7 Suppose that A and B are endomorphisms of a finite-dimensional vector space V over a field K. Prove or disprove the following statements:

1. Every eigenvector of AB is also an eigenvector of BA.

2. Every eigenvalue of AB is also an eigenvalue of BA.

Problem 8 Classify all abelian groups of order 80 up to isomorphism.

Problem 9 Let R be the ring of $n \times n$ matrices over a field. Suppose S is a ring and $h : R \to S$ is a homomorphism. Show that h is either injective or zero.

Problem 10 Let $f : \mathbb { R } \to \mathbb { R }$ be a bounded function $( i . e .$ , there is a constant M such that $| f ( x ) | \leqslant M$ for all $x \in \mathbb { R } )$ . Suppose the graph of f is a closed subset of R2. Prove that f is continuous.

Problem 11 Suppose that $f ^ { \prime \prime } ( x ) = ( x ^ { 2 } - 1 ) f ( x )$ for all $x \in \mathbb { R }$ , and that $f ( 0 ) = 1 , f ^ { \prime } ( 0 ) = 0$ . Show that $f ( x ) \to 0 \ a s \ x \to \infty$

Problem 12 Evaluate the integral

$$
\int _ { - \infty } ^ { \infty } { \frac { \sin ^ { 2 } x } { x ^ { 2 } } } d x .
$$

Problem 13 Suppose that $f : \mathbb { C } \to \mathbb { C }$ is injective and everywhere holomorphic. Prove that there exist $a , b \in \mathbb { C }$ with $a \neq 0$ such that $ f ( z ) = a z + b$ for all $z \in \mathbb { C }$

Problem 14 Show that

$$
\operatorname* { d e t } ( \exp ( M ) ) = e ^ { \operatorname { t r } ( M ) }
$$

for any complex $n \times n$ matrix M, where exp(M) is defined as in Problem ??.

Problem 15 Suppose that P and Q are $n \times n$ matrices such $t h a t P ^ { 2 } = P .$ $Q ^ { 2 } = Q$ , and $1 - ( P + Q )$ is invertible. Show that P and Q have the same rank.

Problem 16 Suppose that A is a commutative algebra with identity over $\mathbb { C } \ \left( i . e . , A \right.$ is a commutative ring containing C as a subring with identity). Suppose further that $a ^ { 2 } \neq 0$ for all nonzero elements $a \in A$ . Show that if the dimension of A as a vector space over C is finite and at least two, then the equations $a ^ { 2 } = a$ is satisfied by at least three distinct elements $a \in A$

Problem 17 Let $G L _ { 2 } ( \mathbb { Z } _ { m } )$ denote the multiplicative group of invertible 2×2 matrices over the ring of integers modulo m. Find the order of $G L _ { 2 } ( \mathbb { Z } _ { p ^ { n } } )$ for each prime p and positive integer n.

Problem 18 Let H be the quotient of an abelian group G by a subgroup K. Prove or disprove each of the following statements:

1. If H is finite cyclic then G is isomorphic to the direct product of H and K.

2. If H is a direct product of infinite cyclic groups then G is isomorphic to the direct product of H and K.