Preliminary Exam - Summer 1978

Problem 1   For each of the following either give an example or else prove that no such example is possible.

1. A nonabelian group.

2. A finite abelian group that is not cyclic.

3. An infinite group with a subgroup of index $5$.

4. Two finite groups that have the same order but are not isomorphic.

5. A group $G$ with a subgroup $H$ that is not normal.

6. A nonabelian group with no normal subgroups except the whole group and the unit element.

7. A group $G$ with a normal subgroup $H$ such that the factor group $G/H$ is not isomorphic to any subgroup of $G$.

8. A group $G$ with a subgroup $H$ which has index $2$ but is not normal.

Problem 2   Let $R$ be the set of 2$\times$2 matrices of the form

$$\left( \begin{array}{cc} a & -b \\ b & a\end{array} \right)$$

where a, b are elements of a given field $\mbox{\bf {F}}$. Show that with the usual matrix operations, $R$ is a commutative ring with identity. For which of the following fields $\mbox{\bf {F}}$ is $R$ a field: $\mbox{\bf {F}} = \mbox{$\mathbb{Q}\,^{}$},\, \mbox{$\mathbb{C}\,^{}$},\, \mbox{$\mathbb{Z}^{}$}_5,\, \mbox{$\mathbb{Z}^{}$}_7$ ?

Problem 3   Let $A$ be a $n\times n$ real matrix.

1. If the sum of each column element of $A$ is $1$ prove that there is a nonzero column vector $x$ such that $Ax = x$.

2. Suppose that $n = 2$ and all entries in $A$ are positive. Prove there is a nonzero column vector y and a number $\lambda>0$ such that $Ay = \lambda y$.

Problem 4  

1. Using only the axioms for a field $\mbox{\bf {F}}$, prove that a system of m homogeneous linear equations in n unknowns with $m<n$ and coefficients in $F$ has a nonzero solution.

2. Use (1) to show that if $V$ is a vector space over $\mbox{\bf {F}}$ which is spanned by a finite number of elements, then every maximal linearly independent subset of $V$ has the same number of elements.

Problem 5   Evaluate

$$\int_0^{2\pi}e^{(e^{i\theta}-i\theta)}\,d\theta\,.$$

Problem 6   Let $f:\mbox{$\mathbb{C}\,^{}$} \to \mbox{$\mathbb{C}\,^{}$}$ be an entire function and let $a>0$ and $b>0$ be constants.

1. If $\vert f(z)\vert \leq a\sqrt{\vert z\vert} + b$ for all z, prove that f is a constant.

2. What can you prove about f if
   
   $$\vert f(z)\vert \leq a\vert z\vert^{5/2} + b$$
   
   
   for all z?

Problem 7  

1. Solve the differential equation $g'=2g$, $g(0)=a$ where $a$ is a real constant.

2. Suppose $f:[0,1] \to \mbox{$\mathbb{R}^{}$}$ is continuous with $f(0)=0$, and for $0<x<1$ $f$ is differentiable and $0 \leq f'(x) \leq 2f(x)$. Prove that $f$ is identically $0$.

Problem 8   Let $\{S_{\alpha}\}$ be a family of connected subsets of $\mathbb{R}^{2}$ all containing the origin. Prove that $\bigcup_{\alpha} S_{\alpha}$ is connected.

Problem 9   Let $X$ and $Y$ be nonempty subsets of a metric space $M$. Define

$$d(X,Y) = \inf \{d(x,y) \;\vert\; x \in X, y \in Y\}.$$

1. Suppose $X$ contains only one point $x$, and $Y$ is closed. Prove
   
   $$d(X,Y) = d(x,y)$$
   
   
   for some
2. Suppose $X$ is compact and $Y$ is closed. Prove
   
   $$d(X,Y) = d(x,y)$$
   
   
   for some $x \in X, \,y \in Y$.
3. Show by example that the conclusion of Part 2 can be false if $X$ and $Y$ are closed but not compact.

Problem 10   Let $U \subset \mbox{$\mathbb{R}^{n}$}$ be a convex open set and $f:U \to \mbox{$\mathbb{R}^{n}$}$ a differentiable function whose partial derivatives are uniformly bounded but not necessarily continuous. Prove that f has a unique continuous extension to the closure of $U$.

Problem 11   Suppose the power series

$$\sum_{n=0}^{\infty}a_nz^n$$

converges for $\vert z\vert<$R where $z$ and the $a_n$ are complex numbers. If $b_n \in \mbox{$\mathbb{C}\,^{}$}$ are such that $\vert b_n\vert<n^2\vert a_n\vert$ for all $n$, prove that

$$\sum_{n=0}^{\infty}b_nz^n$$

converges for $\vert z\vert<$R.

Problem 12  

1. Suppose f is analytic on a connected open set
   $U \subset \mbox{$\mathbb{C}\,^{}$}$ and f takes only real values. Prove that f is constant.

2. Suppose $W \subset \mbox{$\mathbb{C}\,^{}$}$ is open, g is analytic on $W$, and $g'(z) \neq 0$ for all $z \in W$. Show that
   
   $$% latex2html id marker 1065 \{ \Re g(z)+ \Im g(z) \;\vert\; z \in W \} \subset \mbox{$\mathbb{R}^{}$}$$
   
   
   is an open subset of $\mathbb{R}^{}$.

Problem 13   Let $R$ denote the ring of polynomials over a field $\mbox{\bf {F}}$. Let $p_1,\ldots,p_n$ be elements of $R$. Prove that the greatest common divisor of $p_1,\ldots,p_n$ is $1$ if and only if there is an $n\times n$ matrix over $R$ of determinant $1$ whose first row is $(p_1,\ldots,p_n)$.

Problem 14   Let $G$ be a finite multiplicative group of 2$\times$2 integer matrices.

1. Let $A \in G$. What can you prove about

   (i)

   $\det A$?

   (ii)

   the (real or complex) eigenvalues of A?

   (iii)

   the Jordan or Rational Canonical Form of A?

   (iv)

   the order of A?

2. Find all such groups up to isomorphism.

Note: See also Problem .

Problem 15   Let $V$ be a finite-dimensional vector space over an algebraically closed field. A linear operator $T : V \to V$ is called completely reducible if whenever a linear subspace $E \subset V$ is invariant under $T$ (i.e., $T(E) \subset E$), there is a linear subspace $F \subset V$ which is invariant under $T$ and such that $V = E \oplus F$. Prove that $T$ is completely reducible if and only if $V$ has a basis of eigenvectors.

Problem 16  

1. Prove that a linear operator $T:\mbox{$\mathbb{C}\,^{n}$} \to \mbox{$\mathbb{C}\,^{n}$}$ is diagonalizable if for all $\lambda \in \mbox{$\mathbb{C}\,^{}$},\, \ker (T-\lambda I)^n=\ker(T-\lambda I)$, where $I$ is the $n\times n$ identity matrix.

2. Show that $T$ is diagonalizable if $T$ commutes with its conjugate transpose $T^*$ (i.e., $(T^*)_{jk} =\overline{T_{kj}}$).

Problem 17   Let $E$ be the set of functions $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ which are solutions to the differential equation $f'''+f''-2f = 0$.

1. Prove that $E$ is a vector space and find its dimension.
2. Let $E_0\subset E$ be the subspace of solutions $g$ such that $\lim_{t\to\infty}g(t) = 0$. Find $g\in E_0$ such that $g(0)=0$ and $g'(0)=2$.

Problem 18   Let $N$ be a norm on the vector space $\mathbb{R}^{n}$; that is,
$N : \mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$ satisfies _norm

$$\begin{array}{l} N(x) \geq 0 \;\; and \;\; N(x) = 0 \;\; only... ... + N(y), \\ N(\lambda x) = \vert\lambda\vert N(x) \end{array}$$

for all $x, y \in \mbox{$\mathbb{R}^{n}$}$ and $\lambda \in \mbox{$\mathbb{R}^{}$}$.

1. Prove that $N$ is bounded on the unit sphere.

2. Prove that $N$ is continuous.

3. Prove that there exist constants $A>0$ and $B>0$, such that for all
   $x \in \mbox{$\mathbb{R}^{n}$}, A\vert x\vert\leq N(x) \leq B\vert x\vert$.

Problem 19   Let $f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$ be continuous. Suppose that $\mathbb{R}^{}$ contains a countably infinite subset $S$ such that

$$\int_p^qf(x)\,dx = 0$$

if $p$ and $q$ are not in $S$. Prove that $f$ is identically $0$.

Problem 20   Let $M_{n\times n}$ denote the vector space of real n$\times$n matrices. Define a map $f:M_{n\times n}\to M_{n\times n}$ by $f(X)=X^2$. Find the derivative of $f$.