## 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.
• 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.
• 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.
• 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.
• 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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