Math 305: Introduction to Abstract Algebra and Number Theory

Diary

FALL 2017

01:30 - 02:45 pm MW MB 139

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:
• 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.