Course Syllabus

Jump to Today

Quick links:

 

Content: This course provides a rigorous introduction to abstract algebra, including group theory and linear algebra.  The formal prerequisites for Math 55 are minimal, but this class does require a commitment to a demanding course, strong interest in mathematics, and some familiarity with proofs and abstract reasoning. 

Important: Due to the course's place in the first-year undergraduate mathematics experience at Harvard and its role in helping our future math concentrators get to know each other and form a community, Math 55 is only open to first-year Harvard College students. 

Lectures: will be held Mondays, Wednesdays, Fridays from 9:00 to 10:15. Starting Friday September 5, we meet in Northwest B-101.  Students are expected to attend all lectures. Handwritten lecture notes will be provided.

Course staff: 

  • Prof. Denis Auroux (auroux@math.harvard.edu) - office hours Mondays & Wednesdays 11:00-12:30 in SC 539. 
  • Course Assistants: Iz Chen, Russell Georgi, Diego Gonzalez Gauss, Lily Levitsky, Lillian MacArthur, Jinho Park, Christoph von Pezold, Charlie Scheuermann, Ishaan Sinha, Enrico Yao-Bate, Hanming Ye  

CA office hours:

  • Mondays 4:30-6:30pm in SC 412, Ishaan/Hanming
  • Tuesdays 7:30-9:30pm in SC  111, Lily/Lillian
  • Wednesdays 4:20-5:35pm in Emerson 318, Jinho
  • Wednesdays 6:30-8:30pm in SC 412, Enrico/Diego
  • Thursdays 7:30-9:00pm in SC 111, Christoph
  • Fridays 7:00-9:00pm in SC 229, Russell/Iz
  • Sundays 7:30-9:30pm in Harvard Hall 105, Charlie

The CAs will hold weekly office hours, as well as various one-time discussions to review the material or go over specific skills (proof writing, LaTeX, etc.).  Even if you don't need help with specific problems on the current assignment, attending some office hours or sections is recommended -- it gives you an opportunity to review and ask general questions about the course material (or related topics).  It's also a chance to get to know the CAs and your fellow students in smaller groups.

Textbooks: there are two required texts, Axler's Linear Algebra Done Right and Artin's Algebra.  (We will aim to cover most of Axler, and most of chapters 2-10 in Artin).
Other books that may (or may not) be helpful at various points in the course include Dummit and Foote's Abstract Algebra, Halmos' Naive Set Theory , Halmos' Finite-Dimensional Vector Spaces , Fulton and Harris' Representation Theory: A First Course , and Serre's  Linear Representations of Finite Groups .   Electronic versions of all of these books except for Artin and Dummit & Foote are freely available to Harvard students from the Springer-Verlag website (by clicking on the links above).

Homework will be assigned weekly, and is due on Wednesday each week.  Assignments will be posted on this site, and should be submitted electronically. The use of LaTeX is encouraged but not required.
Doing the homework in a timely fashion is essential to learning the material properly; given the pace, it is extremely hard to catch up if you fall behind in this class. 
Extensions will be granted for illness or other serious circumstances (having overcommitted yourself does not count), but should be requested ahead of the deadline; outside of these cases, late submissions will incur a penalty to be computed at the end of the semester once your cumulative lateness across all assignments exceeds 96 hours (4 days).

Exams: there will be two in-class midterms (Wed Oct 1 and Wed Nov 5) and a take-home final exam (posted Mon Dec 8, due Mon Dec 15). 

Course grades will be based on your homework (50%), the two midterms (25%), and the final exam (25%).
One homework score will be dropped for everyone, so you may miss one assignment without penalty, but you are still responsible for working through the material.

Community:  One of the best features of Math 55 is the sense of community that most students get out of it.  Getting to know the CAs and your fellow students early in the semester, and forming study groups, is an important part of the experience -- they'll be your support network when the math gets rough. Drop by office hours to introduce yourself and meet others (even if you don't have any specific questions). Participate in the Slack discussions. And please remember: it is up to you to make this community inclusive, welcoming, and supportive of all of its members -- and of all the other people around you, including those who aren't in Math 55.

Academic integrity policy: You are encouraged to discuss and collaborate with each other on the homework assignments. However, make sure that you can work through the problems yourself, and write up your answers on your own. This is not only a matter of academic integrity, but also crucial for properly learning the material and the problem-solving skills that this course aims to cover. See also the course's AI policy, which prohibits the use of AI tools to find or write up your solutions to homework problems.   For exams, collaboration or consultation of sources other than those explicitly permitted is not allowed.

Homework assignments

Direct links to the homework assignments will be posted here as PDF files. Go to "Assignments" to see the LaTeX source.

List of Topics and Lecture Notes

Direct links to the lecture notes will be posted here as PDF files.

Here is a tentative list of topics / dates; we may deviate from these.

  • Lecture 1 (Wed Sep 3): Course logistics; groups, examples
  • Lecture 2 (Fri Sep 5): Set theory; more examples; subgroups; homomorphisms
  • Lecture 3 (Mon Sep 8): Subgroups of Z; cyclic groups; equivalence relations; cosets
  • Lecture 4 (Wed Sep 10): Cosets, normal subgroups, quotient groups; exact sequences
  • Lecture 5 (Fri Sep 12): Exact sequences; the symmetric group; center, commutators; free groups; rings and fields
  • Lecture 6 (Mon Sep 15): Rings and fields; vector spaces; independence, span, basis
  • Lecture 7 (Wed Sep 17): Bases, dimension; linear maps and matrices; direct sums
  • Lecture 8 (Fri Sep 19): Change of basis; direct sums; rank and dimension formula; quotient spaces
  • Lecture 9 (Mon Sep 22): Quotients, duals and transpose; linear operators, invariant subspaces
  • Lecture 10 (Wed Sep 24): Linear operators; invariant subspaces, eigenvectors, eigenvalues; triangular matrices
  • Lecture 11 (Fri Sep 26): Triangular matrices and eigenvalues; generalized kernel, generalized eigenspaces; nilpotent operators
  • Lecture 12 (Mon Sep 29): Jordan normal form; characteristic and minimal polynomials; real operators
  • Lecture 13 (Wed Oct 1): Midterm 1 (Review sheet)
  • Lecture 14 (Fri Oct 3): Categories and functors (Handout); bilinear forms; orthogonality
  • Lecture 15 (Mon Oct 6): Orthogonality; inner products; orthonormal bases; orthogonal operators
  • Lecture 16 (Wed Oct 8): Orthogonal and self-adjoint operators; spectral theorem for real operators; Hermitian inner products
  • Lecture 17 (Fri Oct 10): Hermitian inner products; the complex spectral theorem; classifying bilinear forms
  • Lecture 18 (Wed Oct 15): Skew-symmetric bilinear forms; tensor product: definition and properties; trace (Handout)
  • Lecture 19 (Fri Oct 17): Trace; tensor algebra; symmetric and exterior algebras; volume and determinant (Handout)
  • Lecture 20 (Mon Oct 20): Volume and determinant; modules over rings
  • Lecture 21 (Wed Oct 22): Classification of finitely generated abelian groups; group actions
  • Lecture 22 (Fri Oct 24): Group actions, orbits and stabilizers; Burnside's lemma; conjugacy classes
  • Lecture 23 (Mon Oct 27): Finite subgroups of SO(3) and regular polyhedra
  • Lecture 24 (Wed Oct 29): Conjugacy classes; groups of order p2; conjugacy in the symmetric group
  • Lecture 25 (Fri Oct 31): Conjugacy in Sn and An; statement of Sylow theorems
  • Lecture 26 (Mon Nov 3): Statement of Sylow theorems; examples, semi-direct products
  • Lecture 27 (Wed Nov 5): Midterm 2  (Review topics + practice problems)
  • Lecture 28 (Fri Nov 7): Proof of Sylow theorems; groups of order 12
  • Lecture 29 (Mon Nov 10): Group presentations, Cayley graph, normal forms: S_n, braid group
  • Lecture 30 (Wed Nov 12): Presentations continued: SL(2,Z) and PSL(2,Z); representations
  • Lecture 31 (Fri Nov 14): Representations of finite abelian groups; irreducibility; Schur's lemma
  • Lecture 32 (Mon Nov 17): Schur's lemma; representations of S_3; symmetric polynomials
  • Lecture 33 (Wed Nov 19): Symmetric polynomials; the character of a representation
  • Lecture 34 (Fri Nov 21): Characters, orthogonality, and consequences; characters of S_4 and A_4
  • Lecture 35 (Mon Nov 24): Irreducible characters form a basis; the representation ring; irreducible characters of S_5 and A_5
  • Lecture 36 (Mon Dec 1): Characters of A_5; restriction and induction, Frobenius reciprocity
  • Lecture 37 (Wed Dec 3): Frobenius reciprocity; real and quaternionic representations
  • Review part 1 (linear algebra) and part 2 (group theory and representations); review problems: from book & extra

Course Summary:

Course Summary
Date Details Due