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The Homomorphism Theorems
We prove the three isomorphism theorems: the first (the image of a homomorphism is isomorphic to the quotient by its kernel), the second (the diamond isomorphism theorem), and the third (cancellation of nested quotients). Each theorem is illustrated with concrete examples and commutative diagrams.
Jordan Normal Form: Beyond Diagonalization
Every complex square matrix is similar to a Jordan normal form — a block-diagonal matrix of Jordan blocks that is unique up to block ordering. We prove existence and uniqueness, connect the Jordan form to the minimal polynomial, and apply it to compute the matrix exponential e^{tA}.
Continued fractions: how to find the best rational approximation to any number
Continued fractions are the Euclidean algorithm in disguise. We develop convergents and the best-approximation property, compute the continued fraction of sqrt(2), connect it to Pell's equation, and sketch the Stern-Brocot tree.
Popular Series
Graph Theory Textbook
From the definition of a graph to planarity, coloring, and matroids. A textbook series building definitions, theorems, and proofs in a systematic progression.
Group Theory Textbook
From the axioms to the classification of finite groups. A textbook series building definitions, theorems, and proofs in a systematic progression.
Linear Algebra Textbook
From the axioms of vector spaces to Jordan normal form. A textbook series building definitions, theorems, and proofs in a systematic progression.
Number Theory Textbook
From divisibility and congruences to p-adic numbers and algebraic integers. A textbook series building definitions, theorems, and proofs in a systematic progression.
Popular Articles
Cosets and Lagrange's Theorem
Beginning with the definition of cosets of a subgroup, we prove Lagrange's theorem --- the assertion that the order of a subgroup divides the order of the group. As applications, we derive Fermat's little theorem and Euler's theorem, and we exhibit the alternating group A_4 as a counterexample to the converse.
Burnside's Lemma in Action: 57 Essentially Different Colorings of a Cube
How many essentially different ways can you paint the six faces of a cube using three colors? Burnside's lemma gives the answer: 57. We derive this by classifying the 24 rotations of the cube and counting fixed colorings for each, building from orbits and stabilizers to the general counting formula.
Direct and Semidirect Products
We develop the two fundamental ways of building new groups from old: the direct product and the semidirect product. After proving the equivalence of internal and external direct products, we define the semidirect product and illustrate it with the dihedral groups, symmetric groups, and groups of order pq.
Group Theory: A Comprehensive Reference
A single-page overview of undergraduate group theory, from the axioms through the Sylow theorems and the structure of finite abelian groups. Includes key definitions, theorems, and proof sketches with a dependency diagram.
Combinatorics meets dynamic programming: when counting becomes computation
Dynamic programming is bottom-up evaluation of a recurrence; a generating function is its analytic closed form. We draw the correspondence between the knapsack DP and generating function multiplication, discuss convolution and FFT, and survey the connections across the entire series.
Number Theory: A Complete Summary of Definitions, Theorems, and Proofs
A single-article survey of undergraduate number theory. Covers divisibility, congruences, the Euclidean algorithm, primes, quadratic residues, arithmetic functions, continued fractions, p-adic valuations, and algebraic integers, together with all key algorithms and a dependency diagram.
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