Messages - Page 2
Correction to the corrections! On Ark 2 problem 19b, the integer n must be bigger than 3.
Geir
Yesterday I did section 10 about cosets and the theorem of Lagrange and the first part of section 11, about direct products. I included an example showing that the converse of Lagrange does not hold. You'll find that example in the book: Example 15.6 on page 146.
Next week I'll do the rest of section 11 about finitely generated abelian groups. I'll probably also have time to do section 13 about homomorphism.
Section 12 is not in the curriculum, but at a certain point we shall speak more about the dihedral groups.
Geir
There where some minor misprints and inaccuracies in Ark2. New version on the web. Exercise 19: b) n must be greater than or equal to 2 Exercise 20: the e should be a 3 Exercise 21: n must be greater than or equal to 3
Exercise 24: The first matrix is not a permutation as it stands! Extra exercise: Why? I have changed it to the one in the book.
Geir
Yesterday, that is Wednesday feb 6, I proved Cayley's theorem (8.16), but I used most of the time on section 9 (Orbits, cycles and the alternating group) and finished that section.
I introduced the sign of a permutation a little differently than the book, but of course the official curriculum is what stands in the book.
We do exercises on friday.
Next wedensday I'l do section 10 (Cosets and Lagrange's theorem), and if there is time, I ll start on section 11 and do direct products.
Geir
NB: About the curriculum: I once said that section 7 (Generating Sets and Cayley Digraphs) was out, but I was a little to quick. What is out is Cayleys Digraphs.
Theorem 7.4 about intersections of subgroups and definition 7.5 about the subgroup generated by a subset are definitely in the curriculum. I spoke about that very early in the course.
Geir
Today I finished section 6 (cyclic groups) and started on section 8 (groups of permutations). I'l do Cayley's theorem and start on section 9 (Orbits, cycles and the Alternating group) next time. We have many exercises on Friday, so I guess there will no time for theory. In case there is, I ll do Cayley's theorem.
Geir
On Friday we did the exercises 1, 2 and 3 on Ark1. I spoke bout cyclic group from the beginning of section 6 and ended up In the middle of theorem 6.14 about subgroups of finite cyclic groups.
On Wednesday I'll finish that, and start on section 8, permutations.
Chapter 7 is not part of the curriculum.
Geir
To day I did section 5 and started on section 6. Friday we shall do exercises, and next Wednesday I 'll continue with section 6 starting with theorem 6.6.
G
Two minor missprints on Ark1:
Exercise 9: The binary operation should be associative. Exrecise 25, ii): There should be a 10 and not a 5 in the exponent.
Corrected version is on the web.
G
Today I looked at the symmetries of a square. Defined a group, showed some of the basic, elementary properties, defined isomorphisms and homomorphisms. Gave a lot of examples of groups. This corresponds to section 4 in the book.
On next wednesday I'll do section 5, subgroups, and may be start on cyclic groups, section 6.
On friday we do the exercises.
G
In todays lecture I spoke about equivalence relations and partitions, congruences mod N, and defined addition and multiplication of congruence classes. This corresponds to the last part of section 0 in the book.
On Friday, I'll do the motivating example (notes are posted) and start on section 4, the definition of groups. I'll mention some of the topics in section 2 and 3, but will not use to much time on it.
Roots of unity wil pop up frequently, so if you are not at ease with that, you should read section 1.
G
I have posted a note about sets and maps. We are going to use sets and maps a lot, so if you are not completely at ease with those concepts, take a look.
G
I have brushed up a small note I made last year about the symmetries of a square, ment as a motivation for the group axioms. I have uploaded a norwegian version - an english version is coming.
Geir
Welcome to MAT2200 !
To begin with, a few practicalities:
The book we shall use is the following:
Fraleigh, John B.: A first course in abstract algebra, 2003. Addison-Wesley. ISBN: 0-321-15608-0, 7th edition.
We shall only use part of it, chapter VIII is not part of the curriculum. The detailed curriculum you will get as the course proceeds, it depends on what we have time to do.
The exercises I will post on the net in seperate "papers" called Ark - the first one is already there! It covers the first three weeks of the course. I'll try to be a few weeks ahead, so in principle each Ark will cover two or three weeks.
I'll update at least every week on what I plan to do and I really did during the lectures. First week WED 16 and FRI 18 I'll do parts of section 0 (equivalence relations and partition, congruences) and give a introductory example about symmetries of a square. Probably we shall time to do section 2 and 3 and may be start...