Math 3140 Fall 21

MATH 3140: Abstract Algebra 1 (Fall 2021)

Syllabus

Final Exam:

Wednesday 12/15, 7:30-10 pm, in class

Schedule

Numbers are for orientation and refer to sections with related material in Goodman, Algebra: Abstract and Concrete.
  1. 08/23: symmetries, multiplication table (1.1-1.3)
  2. 08/25: linear maps, symmetries of the square as matrices (1.4)
  3. 08/27: symmetries of the square as permutations, Sym(X), S_n (1.5)
  4. 08/30: cycle notation of permutations (1.5), groups, matrix groups, permutation groups (1.10)
  5. 09/01: uniqueness of identity, inverses in groups (2.1), modular arithmetic, equivalence relation, Z_n (1.7)
  6. 09/03: symmetries of a regular n-gon, dihedral group D_2n, +,. on Z_n welldefined (1.7)
  7. 09/08: Euclidean algorithm for gcd, Bezout's coefficients, multiplicative inverses in Z_n (1.7)
  8. 09/10: subgroups (2.2), center of a group
  9. 09/13: center of D_2n, cyclic subgroups, generators (2.2)
  10. 09/15: order of groups and elements, homomorphisms (2.4)
  11. 09/17: properties of homomorphisms (2.4), isomorphisms, cyclic subgroup are isomorphic to Z or Z_n (2.2.20)
  12. 09/20: subgroups of cyclic groups (2.2.21, 2.2.30), permutation groups
  13. 09/22: Cayley's Theorem, cosets of subgroups
  14. 09/24: Lagrange's Theorem 2.5.6
  15. 09/27: REVIEW
  16. 09/29: MIDTERM 1
  17. 10/01: discussion of midterm
  18. 10/04: index of subgroups, applications of Lagrange's Theorem
  19. 10/06: groups of order 2p are either cyclic or dihedral, direct products of groups (3.1.1)
  20. 10/08: kernel, image of homomorphisms, conjugacy, normal subgroup (2.4.16)
  21. 10/11: quotient groups 2.7.1
  22. 10/13: homomorphism theorem 2.7.6
  23. 10/15: correspondence theorem 2.7.13
  24. 10/18 internal direct products 3.1, finitely generated abelian groups 3.6
  25. 10/20: Exchange Lemma
  26. 10/22: Fundamental Theorem of finitely generated abelian groups 3.6.21
  27. 10/25: Fundamental Theorem of finitely generated abelian groups (2)
  28. 10/27: Group actions, orbits, transitivity 5.1
  29. 10/29: stabilizers, orbit size (5.1.14)
  30. 11/01: REVIEW [pdf]
  31. 11/03: MIDTERM 2
  32. 11/05: discussion of midterm
  33. 11/08: fixed points, Burnside-Frobenius lemma (5.2.2)
  34. 11/10: counting orbits
  35. 11/12: Rubik's cube in GAP, [pdf], hardness of finding shortest solutions [arxiv], conjugation (5.1.17), class equation (5.4)
  36. 11/15: p-groups (5.4.2), groups of size p^2 (5.4.3)
  37. 11/17: Sylow subgroups (5.4.7), 1st Sylow Theorem
  38. 11/19: 2nd and 3rd Sylow Theorem (5.4.10-11)
  39. 11/29: groups of size pq (5.4.12), rings (6.1)
  40. 12/01: units, fields, subrings, ring homomorphisms (6.1)
  41. 12/03: ideals, principal ideals (6.2), quotient rings (6.3)
  42. 12/06: Homomorphism Theorem for rings, direct products of rings, Chinese Remainder Theorem
  43. 12/08: REVIEW [pdf]

Assignments

  1. due 08/25 [pdf]
  2. due 09/01 [pdf]
  3. due 09/10 [pdf] [tex]
  4. due 09/15 [pdf] [tex]
  5. due 09/22 [pdf] [tex]
  6. due 09/27 [pdf] [tex] Practice problems due Monday!
  7. due 10/06 [pdf] [tex] [solutions]
  8. due 10/13 [pdf] [tex] [solutions]
  9. due 10/20 [pdf] [tex]
  10. due 10/27 [pdf] [tex] [solutions]
  11. due 11/01 [pdf] [tex] Practice problems due Monday, 8 pm!
  12. due 11/10 [pdf] [tex] [solutions]
  13. due 11/17 [pdf] [tex] [solutions]
  14. due 12/03 [pdf] [tex] [solutions]

Handouts

  1. Integers mod n [pdf]
  2. Basic definitions in group theory [pdf]
  3. List of topics [pdf]