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 and b is an integer linear combination of a and b.
- 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.
- 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 D4 and the general
definition of a group.
- Read: Chapter 2 (up to but not including the elementary properties
of groups sections).
- Homework (due 1/31): Posted here.
- Definitions: Dihedral group D4, binary operation,
group.
- Friday, January 24 -
- Class: Examples of groups.
- Read: The rest of Chapter 2.
- Homework (due 1/31): Posted here.
- Definitions: Abelian, Zn, 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: ez.
- 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