# Math 366: Introduction to Abstract Algebra

## Colorado State University, Spring 2021

**Instructor:** Henry Adams
**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*

^{-1}a^{-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 S*

_{n}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 |