Group Theory in Mathematics for Summer 2025
Teacher: Caleb Alons
Email: calebscottalonsmathematics@gmail.com
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 2025 Group Theory in Mathematics syllabus).
TUITION: $250
WHY THIS COURSE?
Most of the math majors I've taught at college have been largely unprepared for the radical shift in difficulty that pure mathematics brings. Rigorous thinking, proof writing, and the axiomatic method of studying mathematics is often hidden away from high school students, which I find a tragedy in the sense that many students get a false conception of what pure mathematics truly is. Many students who struggle to enjoy the strictly computational method and emphasis of typical high school curricula often find the world of abstract mathematics to be far more enjoyable; my original heart for writing the text for this course was to make the beauty of abstract algebra accessible to high school students who are willing to challenge their minds at a deeply abstract and rigorous level. This course is one of a kind for its pedagogical method that breaks down advanced university mathematics in a way so that high school students can access and explore it.
Syllabus for
Group Theory in Mathematics
Summer 2025
I. COURSE DESCRIPTION
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.
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.
II. STUDENT LEARNING OUTCOMES FOR THIS COURSES
A. 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. Proof by 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
B. 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
III. TEXTBOOKS AND OTHER LEARNING RESOURCES
A. REQUIRED MATERIALS
1. Textbooks:
a. Alons, Caleb S. Abstract Algebra: A First Look at the Fundamentals of Group Theory. Unpublished manuscript, June 2023, typescript.
Access to the textbook is provided with the course; a PDF copy will be provided via the course website.
2. Other:
a. Access to the Internet to log onto the course website
B. OPTIONAL MATERIALS
1. Textbooks:
a. 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.
IV. POLICIES AND PROCEDURES
A. 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:
a. Weekly Homework Assignments:
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
Self-grade work with solutions key: 2 points
___________________________________________
Expected Section Total: 10 points
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.
b. Student Engagement:
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.
c. Grading Scale:
A: 189–210 points
B: 168–188 points
C: 147–167 points
D: 126–146 points
F: 0–125 points
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.
V. 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).
Please email calebscottalonsmathematics@gmail.com if you would like a PDF copy of the syllabus.
SUMMER 2025
Week 1 (Jul 14–20): 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 (Jul 21–27): 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 (Jul 28–Aug 3): 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 (Aug 4–10): 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
VI. INSTRUCTOR QUALIFICATIONS
Caleb Scott Alons will receive his B.S. in Mathematics in May 2025. Caleb is a returning AP Homeschoolers instructor, having previously taught two years of supplementary and preparatory summer courses in both mathematics and science. After developing a love for teaching mathematics, Caleb started professionally tutoring mathematics in 2020 and has been an assistant lecturer at Oral Roberts University for the past four years, teaching several hundred students in Calculus I/II/III, Vector Calculus, Differential Equations, Discrete Mathematics, Linear & Matrix Algebra, Abstract Algebra, Real Analysis, General (Point-Set) Topology, and Algebraic Topology. 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