**Math 305: Introduction to Abstract Algebra and Number Theory **

**Diary**

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FALL 2017__

01:30 - 02:45 pm MW MB 139

[Home][Diary][Assignments] [LaTeX links][Exam]

- August 23, 2017:
- Brief discussion of nature of course
- The equation \(x^2 - d y^2 = 1\).
- Definition of a group.

- August 28, 2017:
- Substitute teaching.

- August 30, 2017:
- Substitute teaching.

- September 4, 2017:
- Labor Day.

- September 6, 2017:
- The Euclidean Algorithm and Lame's Theorem.

- September 11, 2017:
- Lame's Theorem.
- Prime numbers.

- 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.

- 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

- 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).

- September 25, 2017:
- Collected Assignment 02.
- Groups, subgroups.
- The symmetric group \((S_n,\circ)\).

- 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 \((C_1, *)\).

- 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.

- 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.

- 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 \(p\).

- October 11, 2017:
- Proof of Cauchy's Theorem.

- October 16, 2017:
- Returned Assignment 04.
- Collected Assignment 05.
- Review for the Midterm.

- October 18, 2017:
- Midterm Examination.

- October 23, 2017:
- Return graded midterm examination.
- Proof of Lagrange's Theorem.

- October 25, 2017:
- Cayley Tables.
- Cosets of a subgroup and Transversals.
- Converting Cayley tables into Sudoku tables.

- 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.

- November 01, 2017:
- Subgroups of \((\textsf{S}_n,\circ)\).

- November 06, 2017:
- Proof of Cayley's Theorem.

- November 08, 2017:
- Even and odd permutations.
- The alternating group (\(\textsf{A}_n,\circ)\) and other subgroups of \((\textsf{S}_n,\circ)\).

- November 13, 2017:
- Any finite cyclic group of order \(n\) is isomorphic to (\(\mathbb{Z}_n,\; + \mod n)\).

- 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.