Class Diary

A summary of what happened in class is posted below, as well as assigned work. You may need a PDF reader to open PDF files posted below.

- Monday, January 13 -
- Class: Syllabus, what the course is about, and the fact that the
gcd of
a andb is an integer linear combination ofa andb . - Read: The sections of Chapter 0 about properties of integers and complex numbers.
- Homework (due 1/24): Fill out the questionnaire here and bring it to Jeremy in his office.
- Definitions: Divides, greatest common divisor.

- Class: Syllabus, what the course is about, and the fact that the
gcd of
- Wednesday, January 15 -
- Class: Modular arithmetic, complex numbers, and the start of discussion of equivalence relations.
- Read: Finish reading Chapter 0 for Friday.
- Homework (due 1/24): Posted here.
- Definitions: Prime number, least common multiple, congruent modulo
n , complex number, equivalence relation.

- Friday, January 17 -
- Class: Equivalence relations and partitions, and functions, together with an example of a function that is (with appropriate modification) one-to-one and onto.
- Read: Chapter 1 for next Wednesday.
- Homework (due 1/24): Posted here.
- Definitions: Equivalence class, partition, function composition, one-to-one, onto.

- Wednesday, January 22 -
- Class: The dihedral group
D and the general definition of a group._{4} - Read: Chapter 2 (up to but not including the elementary properties of groups sections).
- Homework (due 1/31): Posted here.
- Definitions: Dihedral group
D , binary operation, group._{4}

- Class: The dihedral group
- Friday, January 24 -
- Class: Examples of groups.
- Read: The rest of Chapter 2.
- Homework (due 1/31): Posted here.
- Definitions: Abelian,
Z ,_{n}U(n) ,GL(2,R) .

- Monday, January 27 -
- Class: Some more examples of groups, as well as some general properties of groups.
- Read: The first five pages of Chapter 3.
- Homework (due 2/7): Posted here.

- Wednesday, January 29 -
- Class: Arithmetic in groups and orders.
- Read: The rest of Chapter 3.
- Homework (due 2/7): Posted here.
- Definitions: Order of a group, order of an element.

- Friday, January 31 -
- Class: Subgroups and subgroup tests.
- Read: Nothing new.
- Homework (due 2/7): Posted here.
- Definitions: Subgroup.

- Monday, February 3 -
- Class: Finite subgroup test, cyclic subgroups, the center.
- Read: Nothing new. We'll finish Chapter 3 on Wednesday.
- Homework (due 2/14, the day of the first exam): Posted here.
- Definitions: Cyclic subgroup generated by an element, subgroup
generated by a set
S , center.

- Wednesday, February 5 -
- Class: Centralizers, and the fact that in a non-abelian group at most 5/8 of pairs of elements commute.
- Read: Nothing new.
- Homework (due 2/14): Posted here.
- Definitions: Centralizer.

- Friday, February 7 -
- Class: Theorem about |ab| in relation to |a| and |b|.
- Read: Chapter 4, the first five pages.
- Homework: Nothing new.

- Monday, February 10 -
- Class: Start of Chapter 4.
- Read: The rest of Chapter 4.
- Homework (due 2/21): Posted here.
- Definitions: Nothing new.

- Wednesday, February 12 -
- Class: Proof of Theorem 4.2, statement and examples involving Theorem 4.3.
- Read: Nothing new.
- Homework (due 2/21): Posted here.
- Definitions:
e .^{z}

- Friday, February 14 -
- Class: First midterm exam.

- Monday, February 17 -
- Class: Proof of Theorem 4.3 and discussed counting elements of various orders.
- Read: The first seven pages of Chapter 5.
- Homework (due 2/28): Posted here.
- Definitions: The Euler phi function.

- Wednesday, February 19 -
- Class: Went over in-class exam, did example of intersecting cyclic subgroups, and started talking about permutation groups.
- Read: Three more pages of Chapter 5.
- Homework (due 2/28): Posted here.
- Definitions: Permutation and permutation group.

- Friday, February 21 -
- Class: Cycle notation, products of disjoint cycles commute. Stated (but didn't prove) theorem about orders of permutations in cycle notation.
- Read: The rest of Chapter 5 up to page 115. I'll prove Theorem 5.5 in a different way than the book.
- Homework (due 2/28): Posted here.
- Definitions: Disjoint cycles.

- Monday, February 24 -
- Class: Orders of permutations, products of 2-cycles, even and odd permutations, and definition of permutation matrices.
- Read: The first five pages of Chapter 6.
- Homework (due 3/6): Posted here.
- Definitions: Even and odd permutations,
A , permutation matrix.

- Wednesday, February 26 -
- Class: Why every permutation is even or odd. Cups magic trick. Start of isomorphisms.
- Read: The proof of Theorem 6.1 for Friday.
- Homework (due 3/6): Posted here.
- Definitions: Isomorphism.

- Friday, February 28 -
- Class: Examples of isomorphisms, and the first two steps of the proof of Cayley's theorem.
- Read: The statements and proofs of theorems 6.2 and 6.3.
- Homework (due 3/6): Posted here.

- Monday, March 2 -
- Class: Conclusion of the proof of Cayley's theorem, and the statement and proof of Theorem 6.2.
- Read: The rest of Chapter 6.
- Homework (due 3/27): Posted here.

- Wednesday, March 4 -
- Class: Automorphisms.
- Read: Chapter 7, the material about cosets and the proof of Lagrange's theorem.
- Homework (due 3/27): Posted here.
- Definitions: Automorphism, inner automorphism, automorphism group.

- Friday, March 6 -
- Class: Cosets and Lagrange's theorem.
- Read: Chapter 7 up through the end of the proof of Theorem 7.3. (I'll prove Theorem 7.3 in class, but only for the case p = 3.)
- Homework (due 3/27): Posted here.
- Definitions: left coset and right coset.

- Monday, March 23 -
- Class: Although March 23 hasn't happened yet, there are three lecture videos posted on Canvas you can watch.
- Read: Up through the statement and proof of Theorem 7.3.
- Homework (due 4/3): Posted here.
- Definitions: Index,
HK .

- Note: For the rest of the semester, all course materials will be posted on Canvas. (I'm trying to make it so you can get everything in one place.)

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Last modified: Monday, 23-Mar-2020 11:56:35 EDT