Math 305: Introduction to Abstract Algebra and Number Theory
 

Diary

FALL 2017

01:30 - 02:45 pm MW MB 139

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  1. August 23, 2017:
    • Brief discussion of nature of course
    • The equation \(x^2 - d y^2 = 1\).
    • Definition of a group.
  2. August 28, 2017:
    • Substitute teaching.
  3. August 30, 2017:
    • Substitute teaching.
  4. September 4, 2017:
    • Labor Day.
  5. September 6, 2017:
    • The Euclidean Algorithm and Lame's Theorem.
  6. September 11, 2017:
    • Lame's Theorem.
    • Prime numbers.
  7. September 13, 2017:
    • Collected Assignment 01.
    • Prime numbers: Euclid's Prime Number Lemma: Each integer larger than 1 has a prime factor.
    • Prime numbers: There are infinitely many.
  8. September 18, 2017:
    • Prime numbers: If \(p\) is a prime number that divides \(a\cdot b\), then \(p\) divides \(a\) or \(p\) divides \(b\).
    • Prime numbers: The Fundamental Theorem of Arithmetic - Part 1: Existence
  9. September 20, 2017:
    • Returned Assignment 01.
    • Prime numbers: The Fundamental Theorem of Arithmetic - Part 2: Uniqueness.
    • Class Discussion: Elementary Number Theory: p. 19 Number 03, Number 04 (a).
  10. September 25, 2017:
    • Collected Assignment 02.
    • Groups, subgroups.
    • The symmetric group \((S_n,\circ)\).
  11. September 27, 2017:
    • The unit circle centered at the origin, and a group operation on it: \((C_1, *)\).
    • Finite subgroups of the unit circle group \((C_1, *)\).
  12. October 02, 2017:
    • Returned Assignment 02.
    • Collected Assignment 03.
    • The group \((\mathbb{Z}_n,\; + \mod n)\).
    • The group \((U_n,\; * \mod n)\) and Euler's \(\phi\) function.
  13. October 04, 2017: (Pinter, Cyclic groups, and Chapter 13)
    • Subgroups and generators of \((\mathbb{Z}_n,\; + \mod n)\).
    • Subgroups and generators of \((U_n,\; * \mod n)\).
    • Cyclic groups and Gauss' theorem on when \((U_n,\; * \mod n)\) is cyclic.
    • Lagrange's Theorem.
  14. October 09, 2017:
    • Returned Assignment 03
    • Collected Assignment 04
    • Brief discussion of part 3 of Assignment 04.
    • Lagrange's Theorem and a theorem of Cauchy.
    • Cauchy's Theorem: If \(p\) is a prime number dividing the order (=size) \(\vert G\vert\) of the finite group \((G,\odot\)), then this group has a subgroup of order \(p\).
  15. October 11, 2017:
    • Proof of Cauchy's Theorem.
  16. October 16, 2017:
    • Returned Assignment 04.
    • Collected Assignment 05.
    • Review for the Midterm.
  17. October 18, 2017:
    • Midterm Examination.
  18. October 23, 2017:
    • Return graded midterm examination.
    • Proof of Lagrange's Theorem.
  19. October 25, 2017:
    • Cayley Tables.
    • Cosets of a subgroup and Transversals.
    • Converting Cayley tables into Sudoku tables.
  20. October 31, 2017:
    • \((\textsf{S}_n,\circ)\) and disjoint cycle decompositions.
    • The number of distinct factors of \(n!\) and subgroups of \((\textsf{S}_n,\circ)\).
    • Isomorphisms.
    • Statement of Cayley's Theorem.
  21. November 01, 2017:
    • Subgroups of \((\textsf{S}_n,\circ)\).
  22. November 06, 2017:
    • Proof of Cayley's Theorem.
  23. November 08, 2017:
    • Even and odd permutations.
    • The alternating group (\(\textsf{A}_n,\circ)\) and other subgroups of \((\textsf{S}_n,\circ)\).
  24. November 13, 2017:
    • Any finite cyclic group of order \(n\) is isomorphic to (\(\mathbb{Z}_n,\; + \mod n)\).
  25. November 15, 2017:
    • Any subgroup of a finite cyclic group is cyclic.
    • For any divisor \(k\) of the order \(\vert G\vert\) of a finite cyclic group \((G,\odot)\) there is a unique subgroup of order k.