Math2108A/Mathematics 3101A

Abstract Algebra I / Algebraic Structures with Computer Applications

 
Course Outline
 
Notice Board

Topics covered in the class.

  • Sept. 10: Introduction; Sets, mappings.
  • Sept. 12: Mappings, Binary operations, Binary relation.
  • Sept. 17: Bijections, Binary relations, Division Algorithm.
  • Sept. 19: Divisibility, greatest common divisor, Euclidean algorithm.
  • Sept. 24: Unique factorization, congruences.
  • Sept. 26: Solve linear congruence, congruence classes, monoids.
  • Test # 1
  • Oct. 1: Examples of Monoids.
  • Oct. 3: Free monoids and Deterministic Finite Automata.
  • Oct. 8: no class.
  • Oct. 10: regular language.
  • Oct. 15: Groups, examples and properties.
  • Oct. 17: properties, subgroups, cyclic groups.
  • Test #2.
  • Oct. 22: Cyclic groups.
  • Oct. 24: Order of elements, multiplicative subgroup of complex numbers.
  • Oct. 29: permutation groups.
  • Oct. 31: Cosets and Lagrange's Theorem.
  • Test #3.
  • Nov. 5: Converse of Lagrange's Theorem, Euler's Theorem, RSA.
  • Nov. 7: Homomorphism, Isomorphism, and Cayley's Theorem.
  • Nov. 12: Direct product. Normal subgroup.
  • Nov. 14: Normal subgroup, quotient groups, Kernel.
  • Test #4.
  • Nov. 19: Homomorphisms, Fundamental theorem of homomorphisms.
  • Nov. 21: Isomorphism thoerems, Rings.
  • Nov. 26: Ideals and quotient rings
  • Nov. 28: Homomorphism and isomorphism thoerems, Maximal/prime ideals.

    Tutorials

  • Tutotial #1: Chapter 1: 17, 20, 22, 25, 26.
  • Tutorial #2: Chapter 2: 15 (a) (f), 18, 27, 30.
  • Tutorial #3: posted in WebCT.
  • Tutorial #4: Chapter 4: 5, 8, 32, 37, 43; Chapter 5: 2 (a) (i).
  • Tutorial #5: Chapter 6: 1, 5 (a) (b), 7, 11; Chapter 7: 7 (a), 8(a); Chapter 9: 3, 4.
  • Tutorial #6: Chapter 10: 1 (a) (b) (e); Chapter 11: 4, 8, 19, find all homomorphic images of Z_{6}; Chapter 16: 1 (b) (c) (g), 12.

    Suggested Exercises.

  • Chapter 1: 1-29.
  • Chapter 2: 1, 5, 15, 16, 18, 22-31.
  • Chapter 3: 1, 19-23, 24-54,
  • Chapter 4: 1-12, 13-43.
  • Chapter 5: 1-6, 20-27.
  • Chapter 6: 1-6, 11-21.
  • Chapter 7: 7-10.
  • Chapter 9: 1-11, 14-17, 26-28, 46-48, 52.
  • Chapter 10: 1-10
  • Chapter 11: 1-6, 8-13, 19
  • Chapter 16: 1-3, 11-12.