MATH 120 -- Modern Algebra

"We may always depend on it that algebra which cannot be translated into good English and sound common sense, is bad algebra." -- William Kingdon Clifford
"Algebra is generous; she often gives more than is asked of her." -- Jean le Rond D'Alembert

Course syllabus

Office hours

My office hours will be in my office, 380-382X, Mondays and Wednesdays 11am-12pm. Our course assistant Jason Lo will hold office hours in 380-U1, Mondays 1-2pm and Tuesdays 1-3pm.

Exams

The midterm will be held in class on Monday, May 10. This exam will cover all the material through (and including) Section 3.4. No notes or calculators will be allowed. The cumulative final will be held 8:30-11:30 on Tuesday, June 8. Exams will be based on the lectures and the homework.

Writing in major assignment

Details for the WIM assignment can be found here. The assignment is due Friday, June 4, by 5pm, and will be worth 15% of your grade.

Homework

Problems are from Dummit & Foote : Third Edition unless otherwise indicated. Solutions to selected exercises generously provided by Jason Lo. Late homework will not be accepted for any reason, but you may drop your lowest *two* assignments.

Assignment 1, due Wednesday, April 7, by 5pm:
Section 1.1. # 4, 5, 6, 7, 22, 25, 29
Section 1.2. # 1, 2, 5, 13, 18
Section 1.3. # 1, 4, 5, 15, 20
Solutions to selected exercises

Assignment 2, due Wednesday, April 14, by 5pm:
Section 1.6: # 6, 7, 11, 14, 15, 17, 26
Section 1.7: # 4, 6, 17, 21, 23
Section 2.1: # 1, 2, 4, 6, 8, 9, 10(a)
Solutions to selected exercises

Assignment 3, due Wednesday, April 21, by 5pm:
Section 2.2: # 3, 7, 8, 14
Section 2.3: # 2, 5, 17, 21
Section 2.4: # 1, 6, 8, 9, 11, 15
Section 2.5: # 2, 9
Solutions to selected exercises

Assignment 4, due Wednesday, April 28, by 5pm:
Section 3.1: # 1, 3, 9, 11, 12, 14, 24, 33, 36, 41
Section 3.2: # 1, 4, 9, 22, 23
Solutions to selected exercises

Assignment 5, due Wednesday, May 5, by 5pm:
Section 3.2: # 14, 19, 20, 21
Section 3.3: # 1, 2, 8, 9, 10
Section 3.4: # 1, 4, 5, 7, 12
Solutions to selected exercises

Assignment 6, due Friday, May 14, by 5pm
Section 3.5: # 4, 6, 11, 12, 13
Section 4.1: # 2, 3, 5, 6, 9
Solutions to selected exercises

Assignment 7, due Friday, May 21, by 5pm
Section 4.2: # 2, 4, 8, 9, 10, 14
Section 4.3: # 2, 3, 5, 8, 10, 11, 13, 27
Solutions to selected exercises

Assignment 8, due Friday, May 28, by 5pm
Section 4.4: # 1, 2, 6, 7, 18, 19
Section 4.5: # 4, 13, 14, 16, 30, 31
Section 5.1: # 1, 5, 11
Section 5.2: # 1, 2, 3
Solutions to selected exercises

Assignment 9, due Friday, June 4, by 5pm
Section 7.1: # 6, 15
Section 7.2: # 10
Section 7.3: # 1, 2, 6
Section 8.1: # 1, 3, 6, 7
(*) Extra credit: show that every finite group of order n>= 3 admits at least one nontrivial automorphism. Can you extend your proof to infinite groups?

Lectures

March 29: Introduction, definition of a group (Section 1.1)
March 31: Dihedral groups, generators and relations (Section 1.2)
April 2: More on generators and relations, symmetric groups (Section 1.3)
April 5: Fields, matrix groups, group isomorphisms (Sections 1.4 and 1.6. Also, read Section 1.5.)
April 7: Group homomorphisms and isomorphisms (Section 1.6)
April 9: Group actions (Section 1.7)
April 12: Subgroups, centralizers and the center, stabilizers of a group action (Section 2.1 and 2.2)
April 14: Cyclic groups (Section 2.3)
April 16: Subgroups generated by subsets (Section 2.4)
April 19: Lattices of subgroups (Section 2.5)
April 21: Introduction to quotient groups (Section 3.1)
April 23: Cosets and normal subgroups (Section 3.1)
April 26: More on cosets and Lagrange's theorem (Section 3.2)
(See also these notes. This sketches a proof that the left cosets are in bijection with right cosets via the map gH -> Hg^{-1}. Note that the map gH |-> Hg may not be well-defined.)
April 28: Cosets, normalizers, applications of Lagrange's theorem (Section 3.2)
April 30: Isomorphism theorems (Section 3.3)
May 3: Composition series and the Holder program (Section 3.4)
May 5: Alternating group (Section 3.5)
May 7: Group actions (Section 4.1)
May 10: Midterm exam May 12: Groups acting on themselves by left multiplication (Section 4.2)
May 14: Groups acting on themselves by conjugation and the class equation (Section 4.3)
May 17: Automorphisms (Section 4.4)
May 19: Introduction to Sylow theorems (Section 4.5)
May 21: Applications of Sylow theorems (Section 4.6)
May 24: Classification of finitely generated abelian groups (Section 5.2)
(We also talked about an application: the multiplicative group of a finite field is cyclic)
May 26: Introduction to rings -- lots of examples, and definitions of integral domains, division rings, fields (Sections 7.1 and 7.2)
May 28: Ring homomorphisms, ideals, PID's, and Euclidean domains (Sections 7.3, 7.4, 8.1, 8.2)