MATH 43, section 1 -- Algebraic Structures Fall 1998

• Syllabus
  • Course Syllabus
    
• Notes
  • From Algebra to Algebraic Structures
• Assignments
  • Assignment 1
  • Assignment 2: From Gilbert and Gilbert: Section 1.1/1,2,4,5,11-14; Section 1.2/4,7 (Note: E is the set of all even integers) Due: in class, Wednesday, 9/16
  • Assignment 3: From Gilbert and Gilbert: Section 1.5/1 a,c,e, 3 a,c,e,g, 5,6,8,21; Section 2.2/1,5,9,22,23,28. Due: in class, Monday, 10/5
  • Assignment 4: From Gilbert and Gilbert: Section 2.3/1h, 3,9,12,13,24; Section 2.4/1,2,3cd,14,18,19,20. Due: in class, Friday, 10/16
  • Assignment 5: From Gilbert and Gilbert: Section 1.6/5,6; Section 2.5/1,3,9,17,23,24,25,26,42,44. Due: in class, Monday, 10/26
  • Assignment 6: From Gilbert and Gilbert: Section 3.1/1,2,4,5,21,27,43,45,46,47; Section 3.2/3,5,6,22. Due: In class, Wednesday, 11/11
  • Assignment 7: From Gilbert and Gilbert: Section 3.1/26; Section 3.2/11,17,18,25; Section 3.3/2,4,6,7 a,f, 8 a,f, 25. Due: In class, Wednesday, 11/18
  • Assignment 8: From Gilbert and Gilbert: Section 3.2/14; Section 3.4/5,10 (Note: the group G here is the group of complex numbers a + bi with a,b real numbers and a <> 0 or b <> 0 (or both), under the multiplication (a+bi)(c+di) = (ac-bd)+(ad+bc)i. You will find a few examples of products in Example 1 on page 256.),12,13,24,25; Section 3.5/1,2,10. Due: In class, Monday, November 30

• Information and Announcements
  

  The final examination for this class will be given at 2:30 p.m. on Monday, December 14 in our usual class room, HA 412. The examination will be constructed to take about 2 hours if you work steadily. However, you will have the entire 3 hour examination period to work on the exam if you need that much time.

  Topics To Be Covered

  This will be a comprehensive examination -- it will cover all of the material we have studied since the start of the semester. Specifically, this means: \bigskip

  1. Logical connectives and truth tables, quantifiers, etc.
  2. Sets, set operations (intersection, union, complement, difference, Cartesian product) and their properties
  3. Mappings, domain and range, the injective and surjective properties, permutations
  4. Binary operations, identity elements, inverses, commutativity, associativity, key examples like function composition, matrix addition and multiplication, addition and multiplication in Z and Z_n, etc.
  5. Properties of Z, including the Well-Ordering Property, properties of the positive integers, etc.
  6. Proof by mathematical induction
  7. Divisibility in Z, the division algorithm, prime numbers and prime factorizations, gcd's, Euclid's algorithm
  8. Equivalence relations
  9. Congruence mod n on Z and the integers mod n, Z_n.
  10. Groups -- the definition, how to determine whether a set with a given binary operation is a group, key examples.
  11. Subgroups, Theorem 3.10, cyclic subgroups, etc.
  12. Cyclic groups and their structure
  13. Isomorphisms and homomorphisms of groups, kernels

  Suggestions on How to Study

  Start by reviewing your class notes and the text. If there are terms in the list of topics above that look unfamiliar, or for which you cannot give a precise definition, be sure to learn those definitions first! The reason is quite simple: You cannot answer a question about this material if you do not know what the terms involved mean. Then go back over your three exams, problem sets, and the write-ups from your discussion reports. If there was some question you did not get the first time, try to work out a solution now. If necessary, consult the solutions I have distributed for many of the assignments. Then go through the review sheet problems for the three exams again. It will be most helpful if you work out solutions again "from scratch" even if you have done those problems before.

  Proofs to Know

  See review sheets for Exams 2 and 3.

  Review Session

  Saturday, December 12, at 6:00 pm in HA 412.

• Related Links
  • A catalog of small finite groups
  • Two-dimensional regular patterns from Chaim Goodman-Strauss, University of Arkansas

Last modified: December 7, 1998