All reading references refer to Dummit and Foote's Abstract Algebra, 3rd edition, except when explicitly marked otherwise.

For more of a progression through the book and videos at the undergraduate level, please visit Taylor's excellent list of videos on abstract algebra

An excellent graduate course on group theory, taught by Richard Borcherds

Syllabus for Algebra qualifying exam

Jump to: Week 2 · Week 3 · Week 4 · Week 5 · Week 6

Week 1

Reading

On using LaTeX:

Problem 1:

  • Section 1.2
Videos

On using LaTeX:

Problem 1:

Week 2

Reading

On group actions:

  • Section 1.7: (Introduction to) Group actions
  • Section 4.1: Group actions and permutation representations (#1, 2, 3, 4)
  • Section 4.2: Groups acting on themselves by left multiplication (#2)
  • Section 4.3: Groups acting on themselves by conjugation (#4)

On the symmetric group:

  • Section 1.3: Symmetric groups (#3)
Videos

On group actions:

Week 3

Reading

On subgroups:

  • Section 2.1: Definitions and examples
  • Section 2.2: Centralizers and normalizers, stabilizers and kernels (#2, 6)

On group homomorphisms:

  • Section 1.6: (Introduction to) Homomorphisms and isomorphisms (#2, 5, look at exercises 20, 22, 23)
  • Section 3.1: Definitions and examples
  • Section 3.2: More on cosets and Lagrange's theorem (#3)
  • Section 3.3: The isomorphism theorems (#2)

On solvable groups:

  • Section 3.4: Composition series and the Holder program (#4)

On automorphisms:

  • Section 4.4: Automorphisms (#2,5)

On direct products:

  • Section 5.1: Direct products (#4)

On p-groups:

  • Section 6.1: p-groups, nilpotent groups, and solvable groups (#4, 6)
Videos

On subgroups:

On group homomorphisms:

On solvable groups:

On direct products:

On p-groups:

Week 4

Reading

On solvable groups:

  • Section 3.4: Composition series and the Holder program (#1, 3, 4)
  • Section 6.1: p-groups, nilpotent groups, and solvable groups (#1, 3, 4)

On group homomorphisms:

  • Section 3.3: The isomorphism theorems (#1, 2, 6)

On group actions:

  • Section 4.3: Groups acting on themselves by conjugation (#2)
  • Section 4.4: Automorphisms (#2, 6)

On p-groups:

  • Section 6.1: p-groups, nilpotent groups, and solvable groups (#5)

On the commutator subgroup:

  • Section 5.4, Proposition 7 (#5)

On the Sylow theorems:

  • Section 4.5, Theorem 18 (#6)
Videos

On solvable groups:

On the isomorphism theorems:

On group actions:

On p-groups:

On the commutator subgroup:

Week 5

Reading

On the Sylow theorems:

  • Section 4.5: The Sylow Theorems (#1--6)

On subgroups:

  • Section 2.2: Centralizers and normalizers, stabilizers and kernels (#1)

On direct products:

  • Section 5.1: Direct products (#2)
Videos

Please suggest videos if you find any good ones!

Week 6

Reading

On the Sylow theorems:

  • Section 4.5: The Sylow Theorems (#1--6)
Videos

Please suggest videos if you find any good ones!