Math 122: Algebra I: Theory of Groups and Vector Spaces, Fall 2025.

Time: 12-1:15 pm, MW, at Science Center (1 Oxford St), 507.

Office hours:

Annabel's OH: Monday 5:30-7:00pm in Science Center 310

Grace's OH: Tuesday 1:15-3:10pm in Science Center 310

Nickolas's OH: Wednesday 7:30-9:30pm, Science Center 113

Gil'i's OH: Friday 11:00am-12:00pm, Leverett Dining Hall

Vasya's OH: Tuesday 3:10-4:10pm and Thursday 5-6pm in Science Center 310

Problem Sets:

  • PSet 1.
  • PSet 2.
  • PSet 3.
  • PSet 4.
  • PSet 5.
  • PSet 6.
  • PSet 7.
  • PSet 8.
  • PSet 9.

    Practice problems before the midterm:

  • Problems.

    Practice problems before the final:

  • Problems.

    Schedule:

  • 9/3: Motivation, definition of a group, examples and first properties, Lecture 1.
  • 9/8: More examples of groups: integers modulo n, permutations, an example of a presentation by generators and relations, Lecture 2, Handwritten notes.
  • 9/10: More examples of generators/relations presentations, notion of a subgroup, every group is a subgroup of a group of permutations, more examples of subgroups, cyclic subgroups, classification of subgroups of the integers, Lecture 3, Handwritten notes.
  • 9/15: Corollaries of the classification of subgroups of the integers (applications to gcd and lcm), classification of cyclic subgroups, Lecture 4, Handwritten notes.
  • 9/17: Examples of cyclic subgroups, order of an element, homomorphisms of groups, examples of homomorphisms, Lecture 5, Handwritten notes.
  • 9/22: Properties of homomorphisms, isomorphisms, examples of isomorphisms, automorphisms, how to define homomorphisms using presentations by generators and relations, conjugation automorphisms and an example in the symmetric group, Lecture 6, Handwritten notes.
  • 9/24: Isomorphisms preserve orders of elements, image and kernel of a homomorphism (basic properties and examples), the kernel controls non-injectivity of a homomorphism, relation among the sizes of a group, the image of a homomorphism, and its kernel, an interesting homomorphism between symmetric groups, Lecture 7, Handwritten notes.
  • 9/29: Kernels and normal subgroups, equivalence relations, cosets, Lecture 8, Handwritten notes.
  • 10/1: Quotient by a normal subgroup has a group structure, first isomorphism theorem, Lecture 9, Handwritten notes.
  • 10/6: Lagrange's theorem and applications, dihedral groups, Lecture 10, Handwritten notes.
  • 10/8: More about dihedral groups -- generators, relations, number of elements, Lecture 11, Handwritten notes.
  • 10/13: Holiday.
  • 10/15: Defining group via generators and relations I -- rigorous approach, Lecture 12, Handwritten notes.
  • 10/20: Defining group via generators and relations II -- finishing the general approach, example of dihedral groups, Lecture 13, Handwritten notes.
  • 10/22: Action of a group on a set, examples, partition into orbits, Lecture 14, Handwritten notes.
  • 10/27: Examples of orbits for a group action, transitive action, stabilizers, examples, orbit is a quotient of a group by the stabilizer, Lecture 15, Handwritten notes.
  • 10/29: Midterm.
  • 11/3: Counting formulas, Cauchy's theorem as an application, Lecture 16, Handwritten notes.
  • 11/5: First Sylow theorem, Lecture 17, Handwritten notes.
  • 11/10: Second and third Sylow theorems, applications, Lecture 18, Handwritten notes.
  • 11/12: Application of Sylow's theorems -- classification of groups of order pq, Lecture 19, Handwritten notes.
  • 11/17: Rings and fields -- definitions, examples (Z/nZ, Z, Q, R, C, matrices, polynomial rings), Lecture 20, Handwritten notes.
  • 11/19: Roots of polynomials, zero divisors, number of zeroes of a polynomial, monoids and group rings, group algebra, Lecture 21, Handwritten notes.
  • 11/24: Modules over a ring, examples -- Z-modules, vector spaces, Lecture 22, Handwritten notes.
  • 11/26: Holiday.
  • 12/1: More examples of vector spaces, infinitely generated examples, description of modules over F[x], Lecture 23.
  • 12/3: Examples of F[x]-modules, F[x]-submodules, ring as a module over itself, ideals, PID with examples, quotients by ideals and irreducible elements, finitely generated modules over PID, Jordan normal form as an application, Handwritten notes -- Lectures 23 and 24.

    Additional notes:

  • Mapping property of quotient groups.
  • Third isomorphism theorem.