Mercyhurst UniversityMath DeptDr Williams Home

Math 280 Modern Algebra I

Class Resources and Handouts

I'm experimenting with using direct links to these files in Dropbox, so you'll always have the most recent and updated version available here. Feel free to download and comment if you like, and let me know if there are any technical issues with this system on your end.

Proof Review - Logic, Tautologies, Proof Techniques, Tips, and Things to Avoid
Course Notes - Full Semester


Wallpaper Groups

Course Information

Meeting Times: Tuesday/Thursday 3:30-4:45 pm
Location: Hirt 209
Office Hours: Monday 9-10 and 12:30-1:30, Tuesday 1-3, Wednesday 12:30-1:30, Thursday 8-9
Prerequisites: Math 150, Math 265

Course Description

This is the first semester of a year long sequence on the study of algebraic structures. Course topics include the properties of numbers, equivalence relations, groups, rings, fields, direct products, homomorphisms and isomorphisms, and the natural development of various number systems.


On successful completion of the course, students will be able to:

Required Materials

We will be using Contemporary Abstract Algebra, 8th Edition, by Joseph A. Gallian. An older edition of the text would be fine. No other texts or materials are required. You will not be required to bring the text to class, so an electronic version is acceptable.
The book may be available as an inexpensive rental. If you plan to take Modern Algebra II (Math 281), it is highly recommended that you purchase the text, as you will need it for both semesters.

Lecture Notes

Because we'll see a few topics slightly out of order from the text, and because I wanted to include a few extra examples, I've typed my own lecture notes for the full semester (you can find a link at the top of this page). There is more information in these notes than you'll be responsible for on exams, but I hope they are still useful and interesting. I would strongly recommend skimming over the relevant notes for each lecture before class starts, so you'll know what to pay attention to or ask about in class.


You will have several assignment due throughout the semester. You should expect to spend a fair amount of time on each assignment - don't wait until the night before it's due to get started! You are free to work together on your assignments, but everyone must submit their own work, in their own words. If you need an extension on an assignment, please let me know ahead of the due date so the same extension can be offered to the rest of the class.

Some assignments may include problems that you will not be required to turn in. Make sure to work on these problems anyway, as they could always appear on an exam.


We will have two midterm exams and a final exam. The final exam will be cumulative, while the midterm exams will focus on more recent material. Both exams will be based on homework problems and the suggested textbook problems that do not need to be turned in.

Exam Dates:
Midterm Exam: Tuesday, October 1
Midterm Exam: Tuesday, November 12
Final Exam: Thursday, December 12, 3:30-5:30 pm

Final Grades

Your final grade will be calculated as follows: Your letter grade will be determined according to the department grading scale:

Learning Differences

In keeping with college policy, any student with a disability who needs academic accommodations must call Learning Differences Program secretary at 824-3017, to arrange a confidential appointment with the director of the Learning Differences Program during the first week of classes.

Links and Resources

Joseph Gallian's Abstract Algebra Website
Website of our textbook's author, including some useful links and information.

MIT OpenCourseWare Abstract Algebra by Michael Artin
Full online course in abstract algebra, as presented by Dr. Michael Artin. Includes notes, videos, problems, and study material. If you're planning to continue in mathematics, I'd also highly recommend purchasing a copy of Artin's Algebra, a graduate level textbook.

Algebra: Abstract and Concrete, Frederick M. Goodman
Free abstract algebra textbook, along with accompanying Mathematica programs and demos from the University of Iowa. Great introduction to symmetry.

Abstract Algebra: Theory & Applications, Thomas Judson
" open-source textbook written by Tom Judson that is designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications."

Harvard Extension School's Abstract Algebra Course
Resources for an abstract algebra course, including videos, audio files, and problem sets covering most of the topics we'll be learning. Also includes a section on linear algebra.

Book of Proof, Richard Hammack
Free textbook offering a good review of the structure and language of proofs. Also includes a section on relations, functions, and cardinality.

Linear Algebra, Jim Hefferon
Linear algebra textbook, if you need a refresher on systems of equations, vector spaces, etc.

Free Software

Wolfram Alpha (Web Application)
Use it to check your work and visualize graphs. From the makers of Mathematica. A (modestly priced) upgrade is available, but the free version allows unlimited computations without an account.

GAP (Unix, Mac OS, or Windows)
"GAP is a system for computational discrete algebra, with particular emphasis on Computational Group Theory. GAP provides a programming language, a library of thousands of functions implementing algebraic algorithms written in the GAP language as well as large data libraries of algebraic objects. See also the overview and the description of the mathematical capabilities. GAP is used in research and teaching for studying groups and their representations, rings, vector spaces, algebras, combinatorial structures, and more. The system, including source, is distributed freely. You can study and easily modify or extend it for your special use."

Sage (Unix or Mac OS)
An open source mathematics software system. Runs natively on Linux and Mac. Plenty of documentation to help offset the learning curve. Based on Python with plenty of useful packages, and you can contribute!

CoCalc (Web Application)
An online computing environment that was initially based on Sage (and in fact used to be called SageCloud). You can run Python, Sage Worksheets, LaTeX, a full Linux terminal, R, Jupyter notebooks, and much more.