Previous: Spring78
Next: Fall78
Preliminary Exam - Summer 1978
Problem 1
For each of the following either give an example
or else prove that no such example is possible.
- A nonabelian group.
- A finite abelian group that is not cyclic.
- An infinite group with a subgroup of index
.
- Two finite groups that have the same order but
are not isomorphic.
- A group
with a subgroup
that is not normal.
- A nonabelian group with no normal subgroups except
the whole group and the unit element.
- A group
with a normal subgroup
such that the
factor group
is not isomorphic to any subgroup of
.
- A group
with a subgroup
which has index
but
is not normal.
Problem 2
Let
![$R$](img11-3.gif)
be the set of 2
![$\times$](img12-3.gif)
2 matrices of the form
where a, b are elements of a given field
![% latex2html id marker 743
$\mbox{\bf {F}}$](img14-3.gif)
. Show that with
the usual matrix operations,
![$R$](img15-3.gif)
is a commutative ring
with identity. For which of the following fields
![% latex2html id marker 747
$\mbox{\bf {F}}$](img16-3.gif)
is
![$R$](img17-3.gif)
a field:
![% latex2html id marker 759
$\mbox{\bf {F}} = \mbox{$\mathbb{Q}\,^{}$},\, \mbox{$\mathbb{C}\,^{}$},\, \mbox{$\mathbb{Z}^{}$}_5,\, \mbox{$\mathbb{Z}^{}$}_7$](img18-3.gif)
?
Problem 6
Let
![% latex2html id marker 870
$f:\mbox{$\mathbb{C}\,^{}$} \to \mbox{$\mathbb{C}\,^{}$}$](img36-3.gif)
be an entire function and let
![$a>0$](img37-3.gif)
and
![$b>0$](img38-3.gif)
be constants.
- If
for all z, prove
that f is a constant.
- What can you prove about f if
for all z?
Problem 8
Let
![$\{S_{\alpha}\}$](img51-3.gif)
be a family of connected subsets of
![% latex2html id marker 941
$\mathbb{R}^{2}$](img52-3.gif)
all
containing the origin. Prove that
![$\bigcup_{\alpha} S_{\alpha}$](img53-3.gif)
is connected.
Problem 9
Let
![$X$](img54-3.gif)
and
![$Y$](img55-3.gif)
be nonempty subsets of a metric space
![$M$](img56-3.gif)
. Define
- Suppose
contains only one point
, and
is closed. Prove
for some
- Suppose
is compact and
is closed. Prove
for some
.
- Show by example that the conclusion of Part 2 can be false
if
and
are closed but not compact.
Problem 10
Let
![% latex2html id marker 997
$U \subset \mbox{$\mathbb{R}^{n}$}$](img69-3.gif)
be a convex open set and
![% latex2html id marker 1001
$f:U \to \mbox{$\mathbb{R}^{n}$}$](img70-3.gif)
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
![$U$](img71-3.gif)
.
Problem 11
Suppose the power series
converges for
![$\vert z\vert<$](img73-3.gif)
R where
![$z$](img74-3.gif)
and the
![$a_n$](img75-3.gif)
are complex numbers. If
![% latex2html id marker 1033
$b_n \in \mbox{$\mathbb{C}\,^{}$}$](img76-3.gif)
are such that
![$\vert b_n\vert<n^2\vert a_n\vert$](img77-3.gif)
for all
![$n$](img78-3.gif)
, prove that
converges for
![$\vert z\vert<$](img80-3.gif)
R.
Problem 12
- Suppose f is analytic on a connected open set
and f takes only real values. Prove that f is constant.
- Suppose
is open, g is analytic on
, and
for all
. Show that
is an open subset of
.
Problem 13
Let
![$R$](img88-2.gif)
denote the ring of polynomials over a field
![% latex2html id marker 1110
$\mbox{\bf {F}}$](img89-2.gif)
. Let
![$p_1,\ldots,p_n$](img90-2.gif)
be elements of
![$R$](img91-2.gif)
. Prove that the greatest
common divisor of
![$p_1,\ldots,p_n$](img92-2.gif)
is
![$1$](img93-2.gif)
if and only if there
is an
![$n\times n$](img94-2.gif)
matrix over
![$R$](img95-2.gif)
of determinant
![$1$](img96-2.gif)
whose first
row is
![$(p_1,\ldots,p_n)$](img97-2.gif)
.
Problem 14
Let
![$G$](img98-2.gif)
be a finite multiplicative group of 2
![$\times$](img99-2.gif)
2 integer matrices.
- Let
. What can you prove about
- (i)
?
- (ii)
- the (real or complex) eigenvalues of A?
- (iii)
- the Jordan or Rational Canonical Form of A?
- (iv)
- the order of A?
- Find all such groups up to isomorphism.
Note: See also Problem
.
Problem 15
Let
![$V$](img102-2.gif)
be a finite-dimensional vector space over an algebraically
closed field. A linear operator
![$T : V \to V$](img103-1.gif)
is called completely
reducible if whenever a linear subspace
![$E \subset V$](img104-1.gif)
is invariant
under
![$T$](img105-1.gif)
(i.e.,
![$T(E) \subset E$](img106-1.gif)
), there is a linear subspace
![$F \subset V$](img107-1.gif)
which is invariant under
![$T$](img108-1.gif)
and such that
![$V = E \oplus F$](img109-1.gif)
.
Prove that
![$T$](img110-1.gif)
is completely reducible if and only if
![$V$](img111-1.gif)
has a basis of
eigenvectors.
Problem 18
Let
![$N$](img130-1.gif)
be a
norm on the vector space
![% latex2html id marker 1322
$\mathbb{R}^{n}$](img131-1.gif)
; that is,
![% latex2html id marker 1328
$N : \mbox{$\mathbb{R}^{n}$} \to \mbox{$\mathbb{R}^{}$}$](img132-1.gif)
satisfies _norm
for all
![% latex2html id marker 1332
$x, y \in \mbox{$\mathbb{R}^{n}$}$](img134-1.gif)
and
![% latex2html id marker 1336
$\lambda \in \mbox{$\mathbb{R}^{}$}$](img135-1.gif)
.
- Prove that
is bounded on the unit sphere.
- Prove that
is continuous.
- Prove that there exist constants
and
, such that for
all
.
Problem 19
Let
![% latex2html id marker 1398
$f:\mbox{$\mathbb{R}^{}$}\to\mbox{$\mathbb{R}^{}$}$](img141.gif)
be continuous. Suppose that
![% latex2html id marker 1400
$\mathbb{R}^{}$](img142.gif)
contains a
countably infinite subset
![$S$](img143.gif)
such that
if
![$p$](img145.gif)
and
![$q$](img146.gif)
are not in
![$S$](img147.gif)
. Prove that
![$f$](img148.gif)
is identically
![$0$](img149.gif)
.
Problem 20
Let
![$M_{n\times n}$](img150.gif)
denote the vector space of real n
![$\times$](img151.gif)
n matrices.
Define a map
![$f:M_{n\times n}\to M_{n\times n}$](img152.gif)
by
![$f(X)=X^2$](img153.gif)
.
Find the derivative of
![$f$](img154.gif)
.
Previous: Spring78
Next: Fall78
Paulo Ney de Souza & Jorge-Nuno Silva
2000-08-10