We will not be using a textbook this semester, but rather a task-sequence adopted for IBL. The task-sequence that we are using was written by me. Any errors in the notes are no one’s fault but my own. In this vein, if you think you see an error, please inform me, so that it can be remedied. I reserve the right to modify them as we go, but I will always inform you of any significant changes. The notes will be released incrementally. Each link below is to a PDF file.

- An Inquiry-Based Approach to Abstract Algebra (complete set of notes)
- Title Page
- Chapter 1: Introduction
- 1.1 What is Abstract Algebra?
- 1.2 An Inquiry-Based Approach
- 1.3 Rules of the Game
- 1.4 Structure of the Notes
- 1.5 Some Minimal Guidance

- Chapter 2: An Intuitive Approach to Groups
- Chapter 3: Cayley Diagrams
- Chapter 4: An Introduction to Subgroups and Isomorphisms
- 4.1 Subgroups
- 4.2 Isomorphisms

- Chapter 5: A Formal Approach to Groups
- 5.1 Binary Operations
- 5.2 Groups
- 5.3 Group Tables
- 5.4 Revisiting Cayley Diagrams and Our Original Definition of a Group
- 5.5 Revisiting Subgroups
- 5.6 Revisiting Isomorphisms

- Chapter 6: Families of Groups
- 6.1 Cyclic Groups
- 6.2 Dihedral Groups
- 6.3 Symmetric Groups
- 6.4 Alternating Groups

- Chapter 7: Cosets, Lagrange’s Theorem, and Normal Subgroups
- 7.1 Cosets
- 7.2 Lagrange’s Theorem
- 7.3 Normal Subgroups

- Chapter 8: Products and Quotients of Groups
- 8.1 Products of Groups
- 8.2 Quotients of Groups

- Chapter 9: Homomorphisms and the Isomorphism Theorems
- 9.1 Homomorphisms
- 9.2 The Isomorphism Theorems

- Chapter 10: An Introduction to Rings
- 10.1 Definitions and Examples
- 10.2 Ring Homomorphisms
- 10.3 Ideals and Quotient Rings
- 10.4 Maximal and Prime Ideals

- Appendix A: Prerequisites
- A.1 Basic Set Theory
- A.2 Relations
- A.3 Partitions
- A.4 Functions
- A.5 Induction

- Appendix B: Elements of Style for Proofs
- Appendix C: Fancy Mathematical Terms
- Appendix D: Definitions in Mathematics

I will not be covering every detail of the notes and the only way to achieve a sufficient understanding of the material is to be digesting the reading in a meaningful way. You should be seeking clarification about the content of the notes whenever necessary by asking questions. Here’s one of my favorite quotes about reading mathematics.

Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?

- Student Contract
- Setting the Stage (first day activity)

Below are links to each exam.

- Exam 1 (take-home) (PDF). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due Monday, October 3)
- Exam 2 (take-home) (PDF). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. For the last question, you will need the following pictures of quilts that correspond to group tables for groups of order 8. (Due Monday, October 31 by 4pm)
- Exam 3 (take-home) (PDF). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due Tuesday, December 6 by 5pm)
- Final Exam (take-home) (PDF). If you are interested in using LaTeX to type up your solutions, contact me and I will send you a link to the source file of the exam. (Due by 5pm on Thursday, December 15)

Mathematics & Teaching

Northern Arizona University

Flagstaff, AZ

Website

928.523.6852

Twitter

Instagram

Facebook

Strava

GitHub

arXiv

ResearchGate

LinkedIn

Mendeley

Google Scholar

Impact Story

ORCID

MAT 226: Discrete Math

MAT 690: CGT

This website was created using GitHub Pages and Jekyll together with Twitter Bootstrap.

Unless stated otherwise, content on this site is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.

The views expressed on this site are my own and are not necessarily shared by my employer Northern Arizona University.

The source code is on GitHub.

Flagstaff and NAU sit at the base of the San Francisco Peaks, on homelands sacred to Native Americans throughout the region. The Peaks, which includes Humphreys Peak (12,633 feet), the highest point in Arizona, have religious significance to several Native American tribes. In particular, the Peaks form the Diné (Navajo) sacred mountain of the west, called Dook'o'oosłííd, which means "the summit that never melts". The Hopi name for the Peaks is Nuva'tukya'ovi, which translates to "place-of-snow-on-the-very-top". The land in the area surrounding Flagstaff is the ancestral homeland of the Hopi, Ndee/Nnēē (Western Apache), Yavapai, A:shiwi (Zuni Pueblo), and Diné (Navajo). We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.