Group Theory in Mathematics

Summer 2026

Teacher: Caleb Alons

Email: calebscottalonsmathematics@gmail.com

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COURSE OVERVIEW:

The syllabus for this 0.5 credit course is provided below (email calebscottalonsmathematics@gmail.com to request a PDF copy of the Summer 2026 Group Theory in Mathematics syllabus).

TUITION: $200

WHY THIS COURSE?

In many ways, the traditional high school mathematics track is a convenient white lie about what mathematics truly is, and in Group Theory in Mathematics, students will have the chance to "look under the hood" of all the mathematics they thought they knew. Group theory is my personal overall favorite: favorite to study, favorite to research, favorite to teach. I love group theory so much that it bothered me that high school students are often never given the opportunity to even hear about it (much less take a fun course on it), and so I wrote a textbook of my own with a target audience of middle and high school students. Students will encounter the beautiful world of pure mathematics in a whole new way in this course, and I have designed this course to be as accessible as possible with as few of prerequisites as possible. Each lesson is written in a conversational and easy-to-read style, and every exercise receives a full-length solution or proof (a feature completely unique to my text). I am elated to continue offering this course with AP Homeschoolers, and I always look forward to introducing my students to the universe of mathematics that has always surrounded them but that they never knew existed.

Syllabus for

Group Theory in Mathematics

Summer 2026

COURSE DESCRIPTION
1. Group Theory in Mathematics is a college-level course designed to expose accelerated mathematics students to the type of mathematics prevalent in higher STEM studies. Group theory is a central pillar of abstract algebra (sometimes called modern algebra and not to be confused with high school algebra courses), and the definitions and concepts of group theory underpin much of pure mathematics and theoretical physics. This course surveys the philosophy of proof-based "axiomatic" mathematics and provides an elementary—yet fully rigorous—introduction to the most important concepts of set theory, binary operations, groups, and group theory. Most students should expect to spend 1–2 hours every weekday to keep up with the pace of this course.
2. Prerequisite(s): The technical bare minimum in terms of prerequisites is Prealgebra, Algebra 1, and some basic Euclidean Geometry. Experience with Algebra 2, Precalculus, and beyond can be found helpful with respect to mathematical and mental maturity but is not mandatory to fully grasp and progress through the course.
STUDENT LEARNAING OUTCOMES FOR THIS COURSE
1. COURSE OUTCOMES
  - After completing this course successfully, students will have learned the following concepts from abstract algebra:
    1. The Nature of Abstract Mathematics
    2. Proof Elements
    3. The Terms Definition, Axiom, and Theorem
    4. Mathematical Induction
    5. Sets and Subsets
    6. Cartesian Products
    7. Functions
    8. Injections and Surjections
    9. Equivalence Relations
    10. Congruence Modulo n
    11. Binary Operations
    12. Closure
    13. Commutative and Associative Operations
    14. Binary Operation Tables
    15. Binary Algebraic Structures
    16. Identities
    17. Groups
    18. Cancellation Laws
    19. Linear Equations
    20. Group Tables
    21. Permutation Groups
    22. Cyclic Groups
2. GENERAL OUTCOMES
  - Students will develop the following skills that do not necessarily have restricted application to this course specifically:
    1. Rigorous mathematical proof writing
    2. Advanced creative problem solving
    3. Study discipline and mental toughness
    4. Seeking help and offering help to peers
    5. Collaborative learning
TEXTBOOKS AND OTHER LEARNING RESOURCES
1. REQUIRED MATERIALS
  1. Textbooks:
    1. Alons, Caleb S. Abstract Algebra: A First Look at the Fundamentals of Group Theory. Unpublished manuscript, June 2023, typescript.
    2. Access to the textbook is provided with the course; a PDF copy will be provided via the course website.
  2. Other:
    1. Access to the Internet to log onto the course website
2. OPTIONAL MATERIALS
  1. Textbooks:
    1. Fraleigh, John B. A First Course in Abstract Algebra, 6th ed. Boston, MA: Addison-Wesley, 1998. ISBN-13: 978-0201335965. (This text is widely considered to be a classic and is a highly recommended text for any student pursuing pure mathematics or STEM fields dependent upon a firm foundation of group theory, rings, and field theory.)
POLICIES AND PROCEDURES
1. COURSE POLICIES AND PROCEDURES
  1. Attendance: Group Theory in Mathematics does not have a live component. Instead, students will engage on the class website throughout the week with the instructor and their fellow peers as they progress through the course material. Students desiring additional help from the instructor need only reach out to the instructor via email.
  2. Evaluation Procedures:
    1. Weekly Homework Assignments:
    2. Sections from the textbook are assigned each week. For each section, the student must do the following: read the textbook section and take notes (4 points), complete the exercises at the end of the section (4 points), and self-grade work with the provided solution key (2 points). The expected section total is 10 points.
    3. Each item listed above is based on completion. Students will report their own completion scores per section by the Sunday of each week before 11:59 PM (EST). If a section does not have exercises, students should simply reward themselves with the points automatically to their section score. Since there are 21 sections, the expected total is 210 points for the entire course.
    4. Student Engagement:
    5. Student engagement and peer collaboration is STRONGLY encouraged in this course. Students will have access to the course website to post questions and to help answer the questions of their peers.
    6. Grading Scale:
      1. 189–210 points
      2. 168–188 points
      3. 147–167 points
      4. 126–146 points
      5. 0–125 points
    7. The end goal of this course is for the student to acquire genuine comprehension and mastery of the material, so the final course grade is designed to reflect the student’s ability to learn, willingness to struggle, and desire to overcome deficits and setbacks.
COURSE CALENDAR
- Note on Due Dates:
  - Completion scores for assigned sections will ALWAYS be due by the Sunday of their assigned week at 11:59 PM (EST).

SUMMER 2026

Week 1 (Jun 15–21): Preliminaries and Elementary Set Theory

0.1 What Is Mathematics?
0.2 Elements of Proof
0.3 Mathematical Induction
1.1 Sets and Subsets
1.2 The Cartesian Product
1.3 Functions
Submit completion scores

Week 2 (Jun 22–28): Functions and Introducing Binary Operations

1.4 Injections and Surjections
1.5 Equivalence Relations
1.6 Congruence Modulo n
2.1 Binary Operations
2.2 Closure
Submit completion scores

Week 3 (Jun 29–Jul 5): Algebraic Properties of Binary Operations; Defining the Group

2.3 Commutative and Associative Operations
2.4 Binary Operation Tables
3.1 Binary Algebraic Structures
3.2 Identities
3.3 Groups
Submit completion scores

Week 4 (Jul 6–12): Elementary Group Theory

4.1 Cancellation Laws
4.2 Linear Equations
4.3 Group Tables
4.4 Permutation Groups
4.5 Cyclic Groups
Submit completion scores

INSTRUCTOR QUALIFICATIONS

Caleb Scott Alons holds a B.S. in Mathematics, a B.A. in Biblical Literature, and is currently pursuing a Ph.D. in Mathematics. Caleb launched the AP Homeschoolers summer mathematics program four years ago and has been teaching with AP Homeschoolers ever since. After developing a love for teaching mathematics as a senior in high school, Caleb started professionally tutoring mathematics in 2020 and has since served hundreds of students in primary mathematics, Algebra I–II, Geometry, Precalculus/Trigonometry, AP Calculus AB/BC, undergraduate Calculus I–III, Differential Equations, Linear & Matrix Algebra, Discrete Mathematics, Abstract Algebra, Real Analysis, General Topology, and Algebraic Topology. Caleb also lectured undergraduate mathematics at Oral Roberts University for four years. In his undergrad, Caleb led as president the KME Honors Mathematical Society and Association of Computing Machinery in addition to being an active participant in the Mathematical Association of America, presenting award-winning research in MAA sectionals and the MAKO UG Research Conference. Caleb also scored three consecutive years in the William Lowell Putnam Mathematical Competition, which is esteemed as the hardest UG mathematical exam in the world.

As someone who formerly struggled with mathematics in high school himself, Caleb is deeply invested in seeing every student succeed no matter the circumstances and has personally tutored many students with learning disabilities to help them achieve 5's on AP Calculus AB/BC, score above the 90th percentile on the PSAT/SAT/ACT math tests, and rank in the top 5% of their university math courses. Ultimately, teaching mathematics is a source of great joy for Caleb, and he delights in serving each student under his instruction.

The teacher is the servant of his students. – CSA

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