Course contents document · University · generated 2026-07-15
Abstract Algebra
9 core topics
+ 18 prerequisite topics taught
as needed · approximately 7 hours of instruction
including spaced review
How the course runs
An adaptive diagnostic (up to
40 questions) places the student on the course's knowledge
graph — topics already known are credited, and instruction begins exactly
at the learning frontier. Every topic is taught with a worked-example
lesson and auto-graded practice; a topic is mastered at
75%+ and then maintained through spaced reviews on an
expanding schedule. A cumulative quiz follows every 6
lessons. Prerequisite gaps below the course are detected and taught rather
than skipped, so completion certifies the whole tower, not just the top.
Core curriculum
Abstract Algebra
· 9 topics
Binary Operations[M]
A rule combining two elements into one — the raw material of algebra.
Groups & the Axioms[M]
Closure, associativity, identity, inverses.
Order of an Element[H]
How many steps until you cycle back to the identity.
Cyclic Groups & Generators[H]
One element whose powers sweep out the whole group.
Subgroups & Lagrange's Theorem[H]
A subgroup's size always divides the group's size.
Permutation Groups[H]
Shuffles compose; disjoint cycle lengths give the order.
Homomorphisms & Kernels[H]
Structure-preserving maps; the kernel measures collapse.
Rings & Zero Divisors[H]
Two operations at once — and multiplication can misbehave.
Fields & Inverses mod p[H]
Every nonzero element divides: arithmetic works perfectly mod p.
Prerequisite material
— taught
automatically when the diagnostic finds gaps
Arithmetic Foundations· 7 topics
Adding & Subtracting Whole Numbers
Multi-digit addition and subtraction.
Multiplication
Multiplying whole numbers.
Division
Dividing whole numbers.
Order of Operations
Parentheses first, then multiplication/division, then addition/subtraction.
Negative Numbers: Adding & Subtracting
Working with numbers below zero on the number line.
Negative Numbers: Multiplying & Dividing
Sign rules for products and quotients.
Exponents
Repeated multiplication in shorthand.
Expressions & Equations· 1 topics
Evaluating Expressions
Substituting a value for a variable.
Functions & Algebra II· 1 topics
Function Notation & Evaluation
Reading f(x) notation and plugging in inputs.
Math Olympiad: MOEMS & AMC 8· 3 topics
Modular Arithmetic
Working with remainders directly.
Permutations
Ordered arrangements.
Combinations
Unordered selections: n choose k.
Discrete Mathematics· 6 topics
Propositional Logic
T/F values combined with AND, OR, NOT, and IMPLIES.
Set Operations
Union, intersection, difference — and inclusion–exclusion.
Functions Between Finite Sets
Counting maps: every input independently picks an output.