Math 110AH: Announcements and Corrections
Announcement
The first Wednesday meeting at 4:00 will be on 6 October 04. We shall give another induction proof that is mathematically more
difficult than the double induction that will be done in class but will
give the same result as a consequence. The result to be proven is the
first step in proving Bertand's hypothesis that there is always a prime
p between n and 2n for any integer n >1.
Announcement
The 65th Annual William Lowell Putnam Mathematical Competition
will be held Saturday, December 4, 2004.
The Putnam competition is open only to regularly enrolled
undergraduates in colleges and universities of the United States
and Canada who have not yet received a degree. Each contestant works
individually. UCLA will nominate three contestants whose combined
score will be the team score and will determine the team's rank.
In addition, all contestants will get an individual score and rank.
The examination will take place in two periods, 8-11 am and 1-4 pm
on December 4. There will be 6 problems (A-1 to A-6) in the morning
session and 6 problems (B-1 to B-6) in the afternoon session.
Each problem will be graded out of 10 points.
The problems all require some ingenuity and originality. Usually
many of the problems don't require specific knowledge of upper
division work, but are nonetheless difficult.
The median score nationwide is often about 3 out of 120, but in
one recent year the median score was 0.
Some more information is given on the departmental undergraduate
web site.
If you are interested please send your name and email to:
mess@math.ucla.edu (Geoffrey Mess).
This year the Putnam committee consists of Geoffrey Mess,
Terence Tao, Christoph Thiele and Luminita Vese. There will
be some practice problem sessions.
Geoffrey Mess
Announcement
4:00 Wednesday meeting 13 October 04:
We shall discuss further the concept of an equivalence relation. We may
also discuss axioms in algebra. This will supplement the current and
future class
discussions.
Announcement
4:00 Wednesday meeting 20 October 04:
We shall continue discussing
axioms in algebra.
Announcement
HW 3 Problem 7
In (i) it should read m >= 1.
Homework 3 -- Hints
- Problem 5: Let F = Z/pZ where p
is a prime.
First show that F is a domain, i.e., if x and y lie in F and satisfy
xy = 0, then x=0 or y=0.
Next show that any finite domain is a field using that any map
f: X --> X where X is a finite set is one-to-one iff it is onto.
Here is another way to prove this. What are the units in F from
classwork?
For GL(n,F): for each r > 0, if x_1, ..., x_r are linearly
independent vectors in F^n determine how many vectors in F^n are
linearly dependent on these vectors. Use this to determine
how many vectors y there are so that x_1, ..., x_r, y are linearly
independent.
- Problem 6: Use the Chinese Remainder Theorem. The only tricky
thing to show is that an n-tuple of units on the right hand side comes
from a unit on the left hand side. It too uses the Chinese Remainder
Theorem. As for \phi(p^r), count the integers from 1 to p^r that are
NOT relatively prime to p^r.
- Problem 7: Use the Division Algorithm
Announcement
Friday 4 November 2004
We will use the Wednesday 4:00 meeting for a review.
Remarks on Midterm
- You must be able to accurately state the main theorems that we
have done. This includes knowing all hypotheses.
- You must be able to give the main mathematical reason why a
major theorem is true.
- You must be able to give applications or corollaries of the major
theorems.
- You must be able to give examples of the major theorems and of the
definitions that we have covered.
- You must be able to give examples of explicitly described objects,
e.g., groups with certain properties.
- Given a specific group you should know if it is cyclic, abelian
(but not cyclic), or non-abelian and know its order.
Homework 4 -- Hints
- Problem 4a: Write G/H = { a_i H | i \in I} and
H/K = {b_j K | j \in J} (where I and J are indexing sets).
Show that G/K = { a_i b_j K | i \in I and j \in J}.
Note: no finiteness needed -- think partitions.
- Problem 5: if f \in Aut(G) and x \in G, what inner automorphism
is f \theta_x f^{-1}?
Note that an automorphism of a cyclic group must take a generator of
that cyclic group to a generator of the group. What are the generators
of Z/nZ under addition?
- Problem 8: Take an element of the group. What if its order is >
p. Can you construct one of smaller order?
- Problem 9: How many distinct elements in two different subgroups
of order 7 are there?
Note on take home part of the Midterm
- You cannot use Sylow's First Theorem to prove or disprove Problem 4
j.
- The word or in mathematics is
always the inclusive or, i.e., a or b or a and b.
Hints for Take Home Midterm.
Here are some hints for the Take Home Midterm. Each hint is
separately linked so that you need not look at all of them.
Remarks on Take Home Midterm.
- All proofs must be handwritten or typed by you.
- In problem 2 if there is a counterexample in either or both, do
not just say this or these are counterexamples. Give some
explanation.
- In Problem 4, G^(n+1) is therefore the subgroup of G generated by
commutators of the form xyx^(-1)y^(-1) where x and y lie in G^(n) and
inverses of such commutators.
- In problem 4, it is probably wise to show that H is characteristic
in G if and only if for any f \in Aut(G), f(H) is a subset of H. The
proof should be analogous to the normal case (viewing conjugation
as a map).
- In problem 5 homomorphic image should have been defined.
If $f : G --> H$ is a group homomorphism then $f(G) =
im(f)$ is called the homomorphic image of $f$.
The homomorphic image of a group $G$ is the image of $G$ under
some group homomorphism, i.e., a homomorphic image of a group is
the image of a homomorphism from the group.
- NEW COMMENT In problem 9, as John pointed
out, F_S(x) should have
been written as F_x(S) [= F_< x >(S)] to conform to our notation.
But the sum
in the equation is still over ALL x in G.
- NEW COMMENT In problem 5, Homomorphism
theorems are the same as Isomorphism theorems.
We will work on the following Workshop Problems Wednesday
1 December 04 during our extra meeting.
