# Math 366: Introduction to Abstract Algebra

## Colorado State University, Spring 2021

Email: henry dot adams at colostate dot edu

Lectures: MWF 12:00-12:50pm Mountain Time on Zoom
Textbook: Abstract Algebra: Theory and Applications by Thomas W Judson. This book is freely available as a PDF, and you can find more resources at the book webpage. The book has selected exercise hints in the back, which is great!

Overview: This course is a rigorous and proof-based introduction to abstract algebra. Topics covered include sets, integers, polynomials, real and complex numbers, groups, integral domains, and fields.

Syllabus: Here is a course syllabus.

Course notes: Here are Henry's course notes from teaching a prior version of this class in Spring 2020 and 2019.

## Homework

You do not need to do your homework assigments in LaTeX; I include the LaTeX source files for your homework below only if it is convenient for you, and in the spirit of making things publicly available!

Homework 1 (LaTeX Source) is due Monday, February 1.
Homework 2 (LaTeX source) is due Friday, February 19.
Homework 3 (LaTeX source) is due Friday, March 12.
Homework 4 (LaTeX source) is due Friday, March 26.
Homework 5 (LaTeX source) is due Friday, April 9.
Homework 6 (LaTeX source) is due Friday, April 30.
Homework 7 (LaTeX source) is due Wednesday, May 12.

## Videos

Abstract Algebra 1: Introduction to group theory

Abstract Algebra 2: When are two groups considered to be the same?

Abstract Algebra 3: An introduction to sets

Abstract Algebra 4: Subsets

Abstract Algebra 5: Introduction to functions

Abstract Algebra 6: How does the output space affect whether a function is onto or not?

Abstract Algebra 7: Composition of functions

Abstract Algebra 8: Why is the composition of two one-to-one functions one-to-one?

Abstract Algebra 9: Function composition is associative

Abstract Algebra 10: The definition of a group

Abstract Algebra 11: The group of nonzero real numbers under multiplication

Abstract Algebra 12: The integers modulo 5 form a group under addition

Abstract Algebra 13: The identity in a group is unique

Abstract Algebra 14: The inverse of any element in a group is unique

Abstract Algebra 15: The inverse of ab is b-1a-1 (socks shoes property)

Abstract Algebra 16: The cancellation law

Abstract Algebra 17: Subgroups

Abstract Algebra 18: Abelian groups

Abstract Algebra 19: Two examples of groups that are not abelian

Abstract Algebra 20: Cyclic groups and subgroups

Abstract Algebra 21: What are the generators of Z/nZ?

Abstract Algebra 22: What are the generators of Z/10Z?

Abstract Algebra 23: Not every group is a cyclic group!

Abstract Algebra 24: Cyclic groups and subgroups are abelian

Abstract Algebra 25: Permutations and permutation groups

Abstract Algebra 26: Identity and inverses in permutation groups

Abstract Algebra 27: The size of the symmetric group Sn is n!

Abstract Algebra 28: What are of the elements of the symmetric group S?

Abstract Algebra 29: How do you write a permutation in disjoint cycle notation?

Abstract Algebra 30: How do you write a product of permutations in disjoint cycle notation?

Abstract Algebra 31: How do you write a product of permutations in disjoint cycle notation?

Abstract Algebra 32: Any permutation can be written as a product of 2-cycles

Abstract Algebra 33: An introduction to the alternating group

Abstract Algebra 34: We should we care about permutation groups?

Abstract Algebra 35: Lagrange's theorem: a subgroup's size divides evenly into the group's size

Abstract Algebra 36: Cosets

Abstract Algebra 37: Cosets drawn via a picture

Abstract Algebra 38: Properties of cosets

Abstract Algebra 39: A proof of Lagrange's theorem

Abstract Algebra 40: Left cosets need not equal right cosets

Abstract Algebra 41: The converse of Lagrange's theorem is not true

Abstract Algebra 42: Introduction to public key cryptography

Abstract Algebra 43: The RSA public key cryptography algorithm

Abstract Algebra 44: A factoring fact related to the RSA cryptography algorithm

Abstract Algebra 45: Fermat's little theorem

Abstract Algebra 46: Why does the RSA algorithm correctly decrypt your message?

Abstract Algebra 47: Introduction to Isomorphisms

Abstract Algebra 48: Which isomorphism class does this group of size four belong to?

Abstract Algebra 49: Definition of isomorphisms

Abstract Algebra 50: An example isomorphism

Abstract Algebra 51: A bijective map that is not an isomorphism

Abstract Algebra 52: Two infinite groups that are not isomorphic

Abstract Algebra 53: Group properties preserved by isomorphisms

Abstract Algebra 54: Ways to tell that two groups are not isomorphic

Abstract Algebra 55: Homomorphisms

Abstract Algebra 56: Example group homomorphisms

Abstract Algebra 57: Examples and non-examples of homomorphisms

Abstract Algebra 58: Kernels of homomorphisms

Abstract Algebra 59: Analogy between group homomorphisms and linear transformations (linear algebra)

Abstract Algebra 60: Quotient groups

Abstract Algebra 61: Quotient group examples

Abstract Algebra 62: Direct products

Abstract Algebra 63: Orders of elements in direct product groups

Abstract Algebra 64: Fundamental theorem of finite abelian groups, Part I

Abstract Algebra 65: Fundamental theorem of finite abelian groups, Part II

Abstract Algebra 66: Introduction to group actions and the orbit-stabilizer theorem

Abstract Algebra 67: The number of rotational symmetries of a soccer ball

Abstract Algebra 68: Stabilizers

Abstract Algebra 69: Orbits

Abstract Algebra 70: The orbit-stabilizer theorem

Abstract Algebra 71: An introduction to fields

Abstract Algebra 72: The finite field of size p, where p is a prime

Abstract Algebra 73: Two equivalent definitions of fields - one condensed, one expanded

Abstract Algebra 74: In a field F, why is 0 times x equal to 0?

Abstract Algebra 75: In a field F, why does xy=0 imply either x=0 or y=0?

Abstract Algebra 76: Rings

Abstract Algebra 77: Subrings

Abstract Algebra 78: The ring of Gaussian integers

Abstract Algebra 79: The ring of all functions from the reals to the reals

Abstract Algebra 80: Zero divisors

Abstract Algebra 81: The cancellation property in integral domains

Abstract Algebra 82: Straight-edge and compass constructions

Abstract Algebra 83: Impossible straight-edge and compass constructions

Abstract Algebra 84: Field extensions for straight-edge and compass constructions

## Schedule

 Date Book Chapter: Class Topic Remark Jan 20 Course overview [Logistics] Jan 22 Course overview Jan 25 Let's meet each other! Jan 27 Chp 1: Preliminaries Jan 29 Chp 1: Preliminaries Feb 1 Chp 3: Groups Homework due Feb 3 Class got turned into office hours due to technical difficulties Last day to drop or change grading option Feb 5 Chp 3: Groups Feb 8 Chp 3: Groups Feb 10 Chp 3: Groups and Subgroups Feb 12 Chp 4: Cyclic Groups Feb 15 Snow day - class cancelled (Short Video) Feb 17 Chp 4: Cyclic Groups Feb 19 Chp 4: Cyclic Groups Homework due Feb 22 Chp 4: Cyclic Groups Feb 24 Chp 5: Permutation Groups Feb 26 Chp 5: Permutation Groups Mar 1 Review break Mar 3 Review break Mar 5 Review break Mar 8 Chp 6: Cosets and Lagrange's Theorem (Video 63-64), (Video 65-67), (Video 68-69) Mar 10 Chp 6: Cosets and Lagrange's Theorem (Video 58), (Video 59-61), (Video 60-62) Mar 12 Chp 7: Cryptography Homework due Mar 15 Snow day - class cancelled Mar 17 Chp 7: Cryptography Mar 19 Chp 9: Isomorphisms (Video 46-48), (Video 49-51) Mar 22 Chp 9: Isomorphisms (Video 55-56), (Video 52-54) Mar 24 Chp 9: Isomorphisms (Video 56-57), (Direct Products Video 86-89) Mar 26 Chp 11: Homomorphisms (Video 77-80) Homework due Mar 29 Chp 11: Homomorphisms (Video 80-82), (Video 83-85) Mar 31 Chp 11: Homomorphisms Apr 2 Chp 13: Finite Abelian Groups Apr 5 Chp 14: Group Actions Apr 7 Chp 14: Group Actions Apr 9 Chp 16: Group Actions Homework due Spring Break, Apr 12-16 Apr 19 Chp 21: Fields Apr 21 Chp 21: Fields Apr 23 Chp 21: Fields Apr 26 Chp 16: Rings Apr 28 Chp 16: Rings Apr 30 Chp 16: Rings Homework due May 3 Chp 17: Polynomials May 5 Chp 17: Polynomials May 7 Chp 18: Integral Domains Wednesday May 12, Homework due