# Math 433, Applied Algebra, Summer 2009

Harold P.
Boas

- Wednesday, July 1
- The second exam was given, and solutions are available.
- Tuesday, June 30
- We reviewed for the exam to be given tomorrow.
- Monday, June 29
- I updated the list of corrections to the textbook.

In class, we discussed the notion of isomorphism of groups, and we looked at isomorphism classes of groups of small order. The sixteenth quiz was given, and solutions are available.

Here is the assignment.- Exercises 1, 3, and 8 on pages 229–230 in section 5.3.
- Review for Exam 2 to be given on Wednesday, July 1, on Chapters 4 and 5.

- Friday, June 26
- We discussed the method of decoding linear codes via the parity-check matrix, coset leaders, and syndromes. The fifteenth quiz was given, and solutions are available.

The assignment is Exercises 2, 6, and 7 on pages 252–253 in section 5.4. - Thursday, June 25
- I updated the list of corrections to the textbook.

In class, we discussed the ISBN-13 and ISBN-10 check-digit algorithms and started the theory of binary error-detecting and error-correcting codes. The fourteenth quiz was given, and solutions are available.

Here is the assignment.- Exercises 1 and 3 on page 252 in section 5.4
- Exercise 4 on page 219 in section 5.2

- Wednesday, June 24
- We discussed cosets and Lagrange's theorem. The thirteenth quiz was given, and solutions are available.

The assignment is Exercises 1, 3, and 5 on pages 218–219 in section 5.2. - Tuesday, June 23
- We discussed subgroups, cyclic subgroups, the notion of order, and solving equations in groups. The twelfth quiz was given, and solutions are available.

The assignment is Exercises 1, 3, and 5 on page 211 in section 5.1. - Monday, June 22
- We discussed the following concepts: algebras over a field, Boolean rings, and Boolean algebras (which are not algebras in the first sense). The eleventh quiz was given, and solutions are available.

The assignment is Exercises 6, 7, and 11 on pages 196–197 in section 4.4.

I updated the list of corrections to the textbook. - Friday, June 19
- We discussed the concepts of ring, field, and vector space. The tenth quiz was given, and solutions are available.

The assignment is Exercises 3, 5, and 9 on pages 196–197 in section 4.4. Watch out for the typo in Exercise 3(iv): the correct formula for the symmetric difference can be found in Exercise 2.1.4 on page 86. - Thursday, June 18
- We looked at some further examples of groups, particularly matrix groups. Also we discussed the weaker structures of monoids and semigroups. The ninth quiz was given, and solutions are available.

Here is the assignment.- Exercises 3 and 8 on pages 183–184 in section 4.3
- Exercise 1 on page 195 in section 4.4

- Wednesday, June 17
- I updated the list of corrections to the textbook.

In class, we looked at the formal definition of a group, considered some examples, proved that identity elements are unique, and proved that left-hand inverses are automatically two-sided inverses. The eighth quiz was given, and solutions are available.

The assignment is Exercises 1 and 2 on page 183 in section 4.3. - Tuesday, June 16
- We briefly discussed the NPR news story this morning about the confirmation of the discovery of the 47th known Mersenne prime, featured at the GIMPS home page. We used a method different from the one in the book to prove that the notion of sign is well defined for permutations. (We showed that every permutation is a product of adjacent transpositions and that multiplication on the right by an adjacent transposition changes the number of order inversions by one unit.) We also looked at the alternating group A(4).

The seventh quiz was given, and solutions are available.

The assignment is Exercises 4, 5, and 11 on page 168 in section 4.2. - Monday, June 15
- The graded exams were returned. We discussed notation for permutations, multiplication of permutations, cycle decomposition, and the concepts of order and sign.

Here is the assignment.- Exercises 1, 3, and 4 on page 158 in section 4.1.
- Exercise 1 on page 168 in section 4.2.

- Friday, June 12
- The first exam was given, and solutions are available.
- Thursday, June 11
- We reviewed briefly for the exam to be given tomorrow. Also we discussed set theory as a model of Boolean algebra, and we looked at examples of injective, surjective, and bijective functions as a warm-up to next week's study of permutations.
- Wednesday, June 10
- We discussed relations and various properties that they might or might not have: reflexivity, symmetry, antisymmetry, weak antisymmetry, transitivity. Also we talked about equivalence relations and about partial orders.

The sixth quiz was given, and solutions are available.

The assignment is Exercises 1 and 2 on page 115 in section 2.3.

Reminder: the first exam is on Friday. - Tuesday, June 9
- We discussed finite-state automata and looked at some examples.

The fifth quiz was given, and solutions are available.

The assignment is Exercises 1–5 on pages 124–126 in section 2.4.

Reminder: the first exam is on Friday, June 12. - Monday, June 8
- We discussed the RSA public-key cryptosystem and worked an example.

The fourth quiz was given, and solutions are available.

Here is the assignment.- Exercise 4 on page 59 in Section 1.5
- Exercises 1, 12, and 13 on pages 74–76 in section 1.6

- Sunday, June 7
- I updated the list of corrections to the textbook.
- Friday, June 5
- We discussed the solution of Exercises 6 and 7 on page 48 in section 1.4, we proved Fermat's (little) theorem and did some examples, and we discussed Euler's totient function φ(n) and Euler's generalization of Fermat's theorem.

The assignment (from section 1.6) is to do the rest of Exercise 2, Exercises 3 and 5, and the rest of Exercise 6 (all on page 75). - Thursday, June 4
- We discussed Fermat primes and the solution of Exercise 7 on page 35, and we did some examples of solving simultaneous linear congruences (in other words, the so-called Chinese remainder theorem).

The third quiz was given, and solutions are available.

Here is the assignment:- Exercises 6 and 7 on page 48 in section 1.4
- Exercises 2(ii) and 3 on page 59 in section 1.5

- Wednesday, June 3
- We talked about Mersenne primes, we proved that the congruence class [a] mod n has a multiplicative inverse in ℤ
_{n}if and only if gcd(a,n)=1, and we discussed solving linear congruences.

The second quiz was given, and solutions are available.

Here is the assignment:- Exercise 7 on page 35 in section 1.3 (can you adapt the reasoning we used in class to solve Exercise 6?)
- Exercises 3 and 5 on page 48 in section 1.4 (you did part of Exercise 3 in class)
- Exercise 1 parts i, ii, and iii on page 58 in section 1.5

- Tuesday, June 2
- We went over some induction exercises, used the method of strong induction to prove the existence part of the unique factorization theorem for integers, discussed the notion of congruence modulo n, and defined the addition and multiplication operations on ℤ
_{n}. The first quiz was given, and solutions are available.

The assignment is to read the relevant sections of the textbook; to be prepared for a quiz on induction, prime numbers, and factorization; and to work Exercises 4, 5, and 6 on page 35 in section 1.3 and Exercise 1 on page 48 in section 1.4. - Monday, June 1
- We discussed the notion of greatest common divisor (gcd), the Euclidean algorithm, and the theorem that the greatest common divisor of positive integers a and b
equals the smallest positive integer of the form
as+bt,
where
s and t
are integers.

The assignment is to read the relevant sections of the textbook, to be prepared for a quiz on the material covered today in class, and to be prepared to present exercises R and R+6 from section 1.2 on pages 23–24, where R denotes your row number. - Friday, May 29
- I posted three corrections to the first chapter of the textbook.
- Thursday, May 28
- The first-day handout is available. Classes start on Monday, June 1.

These pages are copyright © 2009 by Harold P. Boas. All rights
reserved.