Course Syllabus

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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 familiarity with proofs and abstract reasoning. 

  • Optional review/warm-up problems can be found here, in case you want to review background on sets and functions, develop your problem-solving skills, or just can't wait.  Attempt them on your own, or join the Slack workspace and meet up with other prospective students to work on these collaboratively.

Take a moment to watch the course presentation video, or read on for more information.

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).

Course staff:

  • Prof. Denis Auroux (
  • Dr. Mark Shusterman (Teaching Fellow)
  • Avery Parr (Course Assistant)
  • Alfian Tjandra (Course Assistant)
  • Richard Xu (Course Assistant)
  • Cheng Zhou (Course Assistant)
  • Gaurav Goel  (Volunteer Course Assistant)

Lectures will be held MWF 10:30-11:45am (US Eastern time), on Zoom. Attendance is highly recommended but not mandatory. Handwritten lecture notes will be provided (see in Files) and lectures will be recorded for those who cannot attend live (see in Panopto).

Discussion sections will be held by the CAs/TF weekly  at various times (see here), in order to go over material from lecture and homework.  These are highly recommended for everyone; students who cannot attend lecture are expected to attend at least one discussion.  Even if you don't need help with specific problems on the current assignment, section is a chance to review and ask general questions about course material (or related issues that may not be dealt with in the lecture), including any content not thoroughly covered in the live lecture.  It's also a chance to get to know the CAs and your fellow students in smaller groups and learn their perspectives.

All students in this class should have access to a tablet (with stylus strongly recommended) in order to facilitate mathematical interaction during discussion sections, office hours, and when collaborating on assignments. If you do not have a tablet, this course should be eligible for loaner iPads/styluses; once registered for the class, contact to request equipment.

Homework will be assigned weekly, and is due on Wednesday each week.  Assignments will be posted on this site, and should be submitted electronically via this website.
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 when circumstances genuinely warrant it (poor time management on your part is not normally a sufficient circumstance), but should be requested ahead of the deadline.

Exams: we will have a take-home midterm (to be taken during the week of September 28) and a take-home final exam (to be taken during reading period).

Course grades will be based on your homework (65%), the midterm (10%), 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: Even though online classes can seem impersonal, keep in mind that everyone in this class is part of a community.  Hopefully, you will soon get to know not just the course staff but also your fellow students.  Drop by office hours to introduce yourself and meet others (even if you don't have any specific questions). Participate in the Slack discussions. Form study groups. And please remember: it is up to you to make this community inclusive, welcoming, and supportive of all of its members.

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.  For exams, collaboration or consultation of sources other than those explicitly permitted is not allowed.

Homework assignments

Here are direct links to the homework assignments as PDF files. Go to "Assignments" to see the LaTeX source, and to submit your solutions.

Lecture notes

Here are direct links to the lecture notes as PDF files. Go to "Files" to see them all in a folder.  The videos are in Panopto (restricted access).

  • Lecture 1 (Wed Sep 2:  Course logistics; groups; examples)
  • Lecture 2 (Fri Sep 4: Set theory; subgroups; homomorphisms)
  • Lecture 3 (Wed Sep 9: Equivalence relations; cosets; normal subgroups; quotient group)
  • Lecture 4 (Fri Sep 11: Normal subgroups, exact sequences; permutation groups; free groups)
  • Lecture 5 (Mon Sep 14: Commutators, free groups; rings and fields; vector spaces)
  • Lecture 6 (Wed Sep 16: Vector spaces; independence, span, basis; dimension; linear maps and matrices)
  • Lecture 7 (Fri Sep 18: Linear maps and matrices; direct sums; rank and dimension formula)
  • Lecture 8 (Mon Sep 21: Quotient spaces; duals; linear operators, invariant subspaces)
  • Lecture 9 (Wed Sep 23: Linear operators, invariant subspaces, eigenvectors and eigenvalues)
  • Lecture 10 (Fri Sep 25: generalized eigenvectors, nilpotent operators, Jordan normal form)
  • Lecture 11 (Mon Sep 28: characteristic polynomial; real operators; categories and functors) (handout)
  • Lecture 12 (Wed Sep 30: categories and functors; bilinear forms, inner products)
  • Lecture 13 (Fri Oct 2: inner products; orthonormal bases; orthogonal and self-adjoint operators)
  • Lecture 14 (Mon Oct 5: orthogonal and self-adjoint operators, spectral theorem; Hermitian forms)
  • Lecture 15 (Wed Oct 7: Hermitian inner products, spectral theorem; classification of bilinear forms)
  • Lecture 16 (Fri Oct 9: tensor products: definition and basic properties) (handout)
  • Lecture 17 (Wed Oct 14: trace; tensor algebra; symmetric and exterior algebras)
  • Lecture 18 (Fri Oct 16: exterior algebra, determinant; modules over rings)
  • Lecture 19 (Mon Oct 19: classification of finitely generated abelian groups; group actions, orbits)
  • Lecture 20 (Wed Oct 21: group actions, orbits and stabilizers, Burnside's lemma; conjugacy classes)
  • Lecture 21 (Fri Oct 23: finite subgroups of SO(3) and regular polyhedra)
  • Lecture 22 (Mon Oct 26: conjugacy classes, class equation; groups of order p^2; symmetric group)
  • Lecture 23 (Wed Oct 28: the alternating group)
  • Lecture 24 (Fri Oct 30: statement of Sylow theorems, examples; semi-direct products)
  • Lecture 25 (Mon Nov 2: proof of Sylow theorems; groups of order 12)
  • Lecture 26 (Wed Nov 4: groups of order 12; group presentations, Cayley graph, normal forms)
  • Lecture 27 (Fri Nov 6: presentations and normal forms: symmetric group, braid group, SL(2,Z))
  • Lecture 28 (Mon Nov 9: representations; sub-representations; irreducibility)
  • Lecture 29 (Wed Nov 11: irreducibility; Schur's lemma; representations of S_3)
  • Lecture 30 (Fri Nov 13: symmetric polynomials; character of a representation)
  • Lecture 31 (Mon Nov 16: characters, orthogonality, and consequences; example: S_4)
  • Lecture 32 (Wed Nov 18: A_4; irreducible characters form a basis; the representation ring)
  • Lecture 33 (Fri Nov 20: representation ring; representations of S_5 and A_5; induced representations)
  • Lecture 34 (Mon Nov 23: induced representations; Frobenius reciprocity; real representations)
  • Lecture 35 (Mon Nov 30: the group algebra; real representations, quaternionic representations)
  • Lecture 36 (Wed Dec 2: real and quaternionic representations; wrap-up, final info and advice)
  • Review part 1: Linear Algebra
  • Review Part 2: Group Theory
  • Review problems from the textbooks


Course Summary:

Date Details Due