This webpage compiles a list of videos corresponding to the various
subsections in Dummit and Foote that we will cover.
Comments on videos
Below is a list of comments that will help guide you through the videos.
- Read the book. The lectures/videos alone are not sufficient for this
course.
- You shouldn't watch all of the videos. You need to
find your own path through the material. This means discussing with
others what you found useful and what you didn't find useful during
weekly Zulip discussions.
- The videos are roughly in order of terseness. Gross is a Field's
medalist and if you want to go fast you could watch his videos.
- Collect your questions!I have tried my best to filter for
prerequisites in these videos but I will definitely fail in this regard.
This is exactly what Zulip discussions are for.
- There are a bunch of different speakers in this list following
different orders. I have tried to indicate where they are in their
lecture series with the "L" followed by a number. So for example "L8"
says they are in lecture 8 of their course. Those videos without numbers
are not part of a series.
Who is in these videos?
James Cook -- following Gallian's
Book
Bendict Gross
(Fields Medalist, Harvard) -- following Artin's
book; he can be terse but is brilliant.
Richard Borcherds
(Fields Medalist at UC Berkeley)-- he is free styling; also this is a
graduate course
Dummit and Foote, Chapter 0
This chapter is a little dry. You will need this material to do the homework
problems though. If you know how to compute the gcd of 270 and 192 by hand
and know how to find integers a and b such that a*270+b*192 = gcd(270,192)
you can skip this section. There is also a Khan
Academy on this.
0.0: Basics
See my Fundamentals
of Mathematics Course for a review
0.1: Properties of the Integers
Math Matters: euclid's
algorithm
Cook: divisibility,
gcd, euclid's algorithm
Learn Math Tutorials: how
to find the greatest common divisor using euclidean algorithm
0.3: Integers Modulo n
Dupuy: modular
arithmetic
Cook: modular
arithmetic
Gross L6: modular
arithmetic
Dummit and Foote, Chapter 1
1.1: Definition of a Group
Neumann: classification
of finite simple groups
Numberphile: Monster
group
Cook L1: a
bit of history, definition of a group
Gross L1: Review
of linear algebra. Groups. Examples of groups. Basic properties and
constructions.
1.2: Dihedral Groups
Borcherds: dihedral
groups (prerequisite: conjugacy classes)
1.3: Symmetric Group
Math Sorceror: a
playlist on the symmetric group (broken up, very nice)
Gross (stand alone) the
symmetric group
1.3: Permutations and cycle notation
Cook L8: permutations
and cycle notation
Gross L2: symmetric
group
Gross L22 (Richard Taylor is the sub): symmetric
group it seems the Harvard course was a little out of touch and
covered the material twice.
BONUS EXAMPLES (if you want more intuition for groups):
Asher Auel: platonic
solids
Michael Penn: the
alternating group
1.4: Matrix Groups
(You should maybe just read this chapter, I've included more videos if
you are curious. Don't spend too much time on this.)
Cook: whole
video series on these.
1.5: Quaternion Group
Michael Penn (single video) quaternion
group:
Borcherds L9: quaternions
3Blue1Brown: quaternions1,
quaternions2
1.6: Homomorphisms and Isomorphisms
Cook: L10: homomorphisms
and isomorphisms
Cook: L11: properties
of isomorphisms
Gross L3: isomorphisms
and homomorphisms
Gross L4: Review,
kernels, normality; Examples; Centers and inner autos Notes for
1.7 Group Actions
Cook L18 group
actions
Borcherds L2: Cayley's
Theorem (he defines group actions)
Mathemaniac: Symmetric
Group and Cayley's Theorem
Gross L17 (skip your first time: Orbit
Stabilizer --- needs cosets, needs orthogonal group)
Gross L18 (skip your first time: Orbit
Stabilizer --- requires cosets)
Gross L19 (skip your first time): Symmetries
of the Platonic Solids, Conjugacy Classes
Dummit and Foote, Chapter 2
2.1: Subgroups (see also the videos for 2.5)
MathDoctorBob L2: definition
of a subgroup
Math Matters: subgroups
and the lattice of groups
2.2: Centralizers, Normalizers, Stabilizers, Kernels
Cook L3: center,
centralizer, symmetries, dihedral groups
Gross L5: equivalence
relations; cosets
2.3: Cyclic Groups and Cyclic Subgroups
Cook L9: cyclic
groups and cyclic subgroups part 1
Cook L10: cyclic
groups and cyclic subgroups part 2
Gross L13: review
from their exam (this is only at the very beginning. The rest is
mostly off topic and talks about linear algebra.)
2.4: Subgroups Generated by subsets of a Group
Eliot 724 (has ads): subgroups
has set generated by a group
2.5: Lattice of Subgroups of a Group
Math Matters: subgroups
and lattice of subgroups
Dummit and Foote, Chapter 3
3.1, 3.2: Quotient Groups and Homomorphisms; More on Cosets and
Lagrange's Theorem
Matt Solomone 302.2B: cosets
of a subgroup
Matt Solomone: quotient
groups (prereq: normal subgroups)
Cook L18:
cosets and Lagrange's theorem part 1
Cook L19: cosets
and Lagrange's theorem part 2
Gross L5: cosets,
lagrange's theorem
Gross: L7 cosets
and quotient groups
Borcherds L12: cauchy's
theorem
Bonus: solvable groups
Matt Solomone 302.4B : solvable
groups
Harpreet Bedi: solvable
groups
3.3: The Isomorphism Theorems
Cook: L14: toward
the first isomorphism theorem
Cook: L15: the
first isomorphism theorem
Cook: L21: isomorphism
theorems
Cook: L22: first
isomorphism, linear algebra, boolean group
The Math Sorcerer: second
isomorphism theorem
Michael Penn: third
isomorphism theorem
3.4: Composition Series and the Holder Program
Borcherds L29: jordan-holder
theorem
3.5: Transpositions and the Alternating Group
Cook: L9: even
and odd permutations
Dummit and Foote, Chapter 4
4.1: Group Actions and Permutation Represenatations
Cook: L19: orbit
stabilizer theorem
4.2: Groups Acting on Themselves by Left Multiplication
Borcherds L2: Cayley's
theorem
4.3: Groups Acting on Themselves by Conjugation
Gross L4: review,
kernels, normality; examples; centers and inner autos
Gross L19 (skip your first time): symmetries
of the platonic solids, conjugacy classes, class equation
4.4: Automorphisms
Gross L4: kernels,
normality; examples centers and inner autos
4.5: Sylow Theorems
Cook: L20: sylow
theorems
Gross L20: applications
of the class equation (needs finite fields), sylow theorems
Gross L21: sylow
theorems part 2
Gross L23: sylow
theorems part 3
4.6: Simplicity of A_n
Math Dr Bob: simplicity
of An
Finite Simple Groups
Cook: L21: isomorphisms
theorems, mention of finite simple group classification
Neumann: classification
of finite simple groups
Ring and Field Theory
Gross L8: introduction
to fields